Amumu is a strong jungler picked semi-regularly in tournament games (post-S2 championship finals). His main strength is his teamfight, doing tons of damage and having massive CC. In the early game he can offer strong ganks, especially in middle lane. Amumu is considered weak to counterjungling, but usually he can manage.
Amumu has particularly good synergy with other AoE heavy champions like Orianna, Malphite, Zyra and Karthus, combined with them doing the Wombo Combo, locking down doing massive damage to several champions at once. Because of his slight weakness to counterjungling, you should consider other picks when something like jungle Nunu is on the enemy team. Even then he can be the best pick though, depending on your team composition.
Passive: Cursed Touch
(Innate) Amumu's autoattacks reduce the target's magic resistance by 15 / 25 / 35 for 3 seconds. The debuff doesn't stack but it refreshes with every autoattack.
A decent passive, though quite uninteresting. Another good reason to autoattack people in teamfights and ganks. When killing small golems, you should autoattack the small golem once to increase your damage against him.
Q Skill: Bandage Toss
(Active) Amumu tosses a sticky bandage in a straight line. If it contacts an enemy, Amumu will pull himself to it, dealing magic damage and stunning the target for 1 second.
Cost: Cost: 80 / 90 / 100 / 110 / 120 mana, Cooldown: 16 / 14 / 12 / 10 / 8 seconds, Magic Damage: 80 / 140 / 200 / 260 / 320 (+0.7 per ability power)
Range: 1,100, Projectile speed: 1800
for ganking, fighting, and hopping over walls in the jungle if you don't need the mana for whatever reason. You can use this skill to initiate teamfights, just hit an important target and ultimate. During a teamfight, use this skill to stick to and stun an important target, usually the enemy AD or AP carry. When your carries are significantly stronger than theirs, you can use this skill to peel for them.
When ganking, you can either shoot this skill from the fog of war (from a brush or from behind a wall) or you have to get in a position where hitting this skill is guaranteed/almost guaranteed. Try to walk up to your target from side or from behind before shooting, dodging from very short range is impossible without flash. Nothing ruins ganks more than a premature shot that ends up missing.
W Skill: Despair
(Toggle) While toggled on, Amumu will be surrounded by a small damaging area of tears. Enemies in the area will be dealt a percentage of their maximum health plus a base amount as magic damage each second.
Cost: 8 mana per second, Cooldown: 1 second, Base Magic Damage: 8 / 12 / 16 / 20 / 24, Max Health to Damage Ratio: 1.5 / 1.8 / 2.1 / 2.4 / 2.7% (+1% per 100 ability power)
Radius of AoE: 300
A nice damage skill. Turn this on when killing jungle creeps or enemy champions, unless you want to conserve your mana. It is not very mana expensive at all though. Not much to add, very straightforward skill.
E Skill: Tantrum
(Passive) Amumu takes reduced physical damage from autoattacks and abilities.
Physical Damage Reduction: 2 / 4 / 6 / 8 / 10
(Active) Amumu will make an instantaneous tantrum, dealing magic damage to surrounding units. Additionally, each time Amumu is hit by an autoattack the cooldown on Tantrum's active will be reduced by 0.5 seconds.
Cost: 35 mana, Cooldown: 10 / 9 / 8 / 7 / 6 seconds, Magic Damage: 75 / 100 / 125 / 150 / 175 (+0.5 per ability power)
Radius of AoE: 200
A great damage/sustain skill. The passive significantly decreases the damage you take from both jungle creeps and lane creeps when ganking. The active is your primary way of jungling and also a very nice damage skill in general.
When jungling, just use this skill whenever off cooldown. When killing buffs, pull the big creep a little bit, so that the small guys aren't hit with your Tantrums, it will speed up killing the buff quite a lot.
When ganking, remember that this skill has a short casting animation, so you might want to delay using this skill in favour of autoattacking with red buff, though that is usually not the case. Also remember that you take no damage from lane creeps at higher levels, so try to aggro them to increase your damage output.
When fighting, you generally want to mash your E button to maximize your damage output, unless you want to conserve mana for Bandage Toss and/or ultimate.
R Skill: Curse of the Sad Mummy
(Active): Amumu entangles surrounding enemy units, dealing magic damage and rendering them unable to move or use autoattacks for 2 seconds.
Cost: 100 / 150 / 200 mana, Cooldown: 150 / 130 / 110 seconds, Magic Damage: 150 / 250 / 350 (+0.8 per ability power)
Radius of AoE: 600
A brilliant ganking and fighting skill. After aquiring this skill, you should generally look for a gank, best in middle or bottom lane. When ganking, try to use this skill before you use Bandage Toss, in order to guarantee the hit on your Q. Use your Q first only if you can't get close enough to cast your ultimate.
In fights, this skill is best used to initiate. Use your Q to close distance and lock down an important target, then ultimate to lock down most of their team. This is especially effective if your team has AoE ultimates of their own, most notably Orianna and Malphite, creating an AoE rape known as "WOMBO COMBO".
If the enemy team is the one to jump on your first, walk/Q/flash in a position to hit as many people as you can.
0/21/9 are really the only viable masteries on Amumu. The extra tankiness, reduced damage from jungle creeps, cooldown reduction and movement speed are more useful than anything else you can get in any other tree. The 9 in utility are mostly for the buff duration, which is very valuable if only for the first few levels.
Marks: Armor or Magic Penetration
Glyphs: MRes, Mres per level, flat Cooldown Reduction, flat AP
Quintessences: Movement Speed
For Marks, Armor is especially useful early game, negating a lot of damage from jungle creeps. Magic Penetration is used by some to improve your combat strength in later levels a little bit, but remember it is at the cost of first jungle clear and sustain in jungle.
Armor Seals are absolutely necessary for jungling, as they reduce the damage from jungle creeps.
There are many options for Glyphs. Flat AP speeds up your jungling a bit, both early and midgame. Flat Cooldown Reduction speeds up your jungling a little less, but more significantly it shortens the extremely long cooldown on your ultimate, allowing for more strong ganks. Magic Resistance or Magic Resistance per level do not help with jungling at all, but they can still be used if you will take large amount of magic damage throughout the game (specifically when enemy team runs double AP).
Movement Speed Quintessences speed up your jungle significantly and increase your combat strength a great amount. Do not underestimate the power of post-mortem jogging.
The only (known) skill order used by pro players is
The reasoning is simple. 1 level in W speeds up jungle significantly, especially buffs, so it is taken very early. E is the best skill to max for both killing jungle creeps fast and taking less damage from them. One level in Q is essential for ganking for the gapcloser and stun.
After maxing E, Q is maxed second in order to reduce its cooldown. In teamfights, the ability to stun and jump twice as often is way more valuable than the minute difference in damage from W, so W is maxed last.
Boots+3 are the starting items for obvious reasons: faster jungling, better ganks and sustain from pots. The standard build is as follows.
Philosopher's Stone is the first item built, the HP and mana regen are important for staying in jungle without having to back too often. It also builds into Shurelya's Reverie, which is one of your core items. The gold per 10 is a small bonus added on top of an already great item.
The standard Amumu core build is Shurelya's Reverie, Aegis of Legion and Boots2.
Boots are usually either Mercury Treads or Ninja Tabi depending on how much CC and physical/magical damage enemy team has. Though Sorcerer's shoes are useable for slightly more fighting strength in smaller fights, they are generally not recommended. Mobility boots generally don't help Amumu's ganks enough to justify them. Without an escape skill, Amumu needs all the fighting strength he can get.
In what order to finish your core items is up to debate. Aegis of the Legion gives you the most fighting strength for the least cost, Boots2 speed up your jungling and help your ganks a little bit, Shurelya's Reverie allows you to initiate teamfights and ganks even better. Generally, build Shurelya's first if they are necessary for your team to follow up your initiate effectively. Otherwise, Aegis+boots2 first are better.
Three items outside of your core that can be built after Philosopher's Stone are Heart of Gold, Doran's Shield and Doran's Ring. All of these items delay your core items quite a bit. Heart of Gold allows you to get Randuin's Omen after you finish your core much faster, but it does weaken your early and midgame in the process. Doran's Ring and Doran's Shield both make you significantly stronger early and midgame, but delaying core items is often not worth it. Doran's Ring gives extra manaregen so that you save on manapots, Doran's Shield is particularly good when you want to towerdive some. However, sticking to your core build is a fine choice in all circumstances.
After your core items there are quite a few options.
For extra armor, Frozen Heart or Randuin's Omen are both a fine choice. Randuin's offers more survivability and an amazing active to follow up your ultimate and is generally the preffered choice. However, Frozen Heart aura and Cooldown Reduction are both more useful in spread out, messy teamfights. So if you expect large amount of kiting from both teams, Frozen Heart is better. However, do not get FH if you already have one on your team, that is a bit of a waste.
For magic resistance, a Negatron Cloak a the most cost efficient choice. The items it builds into are better bought when you run out of itemslots. Banshee's Veil is stronger in simple, cluster-y teamfights that are decisively initiated, so that the spellshield can block a meaningful spell. Force of Nature is better when there is a lot of poking and kiting going on, that is when the HP regen and movement speed are the most helpful. Abyssal Scepter is good when you can spare little bit of gold on damage and you have a lot of magic damage on your team, like Karthus or Katarina.
As for pots, Wards and Oracles. HP pots are a necessity in the first jungle clear and a good buy for the 2nd one. After you get Philostone they should be unneeded. Mana pots are a very good buy throughout the early game. They should be used in case of emergency, when you are low on mana but a fight breaks out or an opportunity presents itself. Buy them in advance then. Wards are great and should be bought throughout the game. Place them to scout for ganks and enemy jungler position. The most vision you have on a map, the safer ganking and farming is.
Because Amumu doesn't have an escape, Oracles shouldn't be bought early. Early game, pink Wards should suffice. Later in the game however, after finishing an item or 2, if everyone but you has a habit of dying or you want to attract some extra attention, Oracles is a good buy.
A very specific item build has been popularized by Azingy and it goes like this.
The first 2 items after Boots of Speed are 2 Doran's Rings, followed by Sorcerer's Shoes. This gives you great fighting strength and extremely fast jungle clear. Azingy then build Abyssal Scepter and Sunfire Cape in whatever order, presumably to deal even more damage. However, it should be noted that this itembuilds requires specific teamcomps and great amount of coordination with teammates, and knowledge of both seems to be exclusive to Azingy's head. Rest of the guide shall assume that you are going the tried-and-true standard itembuild.
The most standard jungle path for both sides is Wolves-Blue-Red. Amumu has very good ganks middle lane thanks to a 1100 range gapcloser, so staying close to mid is generally a good idea. For blue side specifically, Wraiths-Red-Blue is a viable path to countergank the expected gank toplane at that time. However, you will need a smiteless leash on red buff to do that.
After killing your buffs, most of your time will be spent clearing small jungle camps. Pushed lanes are good places to gank, so you can try to walk in and get a kill. Make sure to check where wards are though, walking into lane just to be seen from a mile away will likely just slow down your farm.
After you hit level6 (standard benchmark here is about 8 minutes, maybe slightly after) your ganks get way stronger. You should seek gank opportunities more aggressively, so at this point don't get mad at middle lane for taking your Wraiths. If you get a kill in middle or bottom lane and are level5+, you can often take Dragon uncontested.
As mentioned above, there are usually 2 different ways to gank before you get your ultimate.
First and most common option is to just walk into a lane from side or behind, get close to the enemy and when they are too close to dodge, shoot your Bandage Toss. If your lanemate has a stun, you can cut all the walking close parts and just jump in after the enemy has been stunned.
The other option is to throw bandage toss from a brush or from behind a wall to hit an unsuspecting enemy. This can only rarely be pulled off though, since getting unseen into brushes that people are close to is no small challange.
In ganks you should turn on your W and mash your E and autoattack for maximum damage.
Because of Amumu's low fighting strength and no escape skills, you are vulnerable to counterganking. Preferably make sure that enemy jungler isn't close or that you can burst down the enemy laner in time. Otherwise it is a risk, often worth taking, but still a risk.
After you get your ultimate, the run-close gank gets even stronger, since you can guarantee your Bandage Toss will hit by using your ultimate first, for a total 3 seconds of hard CC. Bandaging in gets significantly stronger as well, simply for the extra damage and CC. When you ultimate is up, you can also countergank extremely well by bandaging in from 1100 range and hitting ultimate on all people present.
Dealing with counterjungle:
The most common place to be counterjungled is your red buff. The enemy jungler can do his blue buff and then run straight to your red. If that happens, you will probably need your laners to come help. Buy time for them by damaging the enemy jungler a bit, attack him and E him when he's attacking the buff, run away when he turns to attack you. Hopefully your teammates arrive in time, if not, you will be forced to try smitesteal.
Bigger problem is if enemy jungler started at your red buff. Have your support or solo laner (depending on what side you are) check your red buff at about 1:56 to see if they are starting there. If they are, tell your middle lane and the lane closer to their red buff push. After finishing your blue go straight to their red, the enemy jungler won't be there in time or you will have a much better position.
If the enemy jungler is just stealing your small camps randomly, there isn't much you can do early on. Try to take your camps, especially your Wraiths, right after they spawn. Taking your small creeps slows down the enemy jungler as well, since they are far away from his own jungle, so you aren't as behind as you might think.
Again, as has been mentioned above, your role in teamfight is usually to initiate by Qing in onto an (important) enemy and pressing R to lock them down. Initiate when your teammates will be able to follow up, so they need to be close enough. Do not initiate teamfights when you know or it is likely that the enemy will win, either because they have the numbers advantage or have a large gold advantage.
If the enemy initiates on your, use your Q and flash to position quickly and hit your ultimate on as many people as you can. Do not delay your ultimate too much though - CCing the enemy isn't very useful when only you and the support are still alive.
After you use your ultimate, use your Q to stick to an important target like AD or AP carry. Mash E button and autoattack to do damage. Amumu is a very strong teamfight champion, so this is your time to shine.
Thanks to: Slayer91, ZERG_RUSSIAN, Simberto, Sandster, sylverfyre, DURRHURRDERP and some more for providing info+insight+grammar check. Special thanks to WaveofShadow for suggesting I do this+making the pictures :3
Note: This guide is for the S2 jungle. Once season 3 starts, this guide will be completely useless BUT I will attempt to update it as soon as possible.