Amumu is one of my favourite champs, due to his powerful late game scaling, his exceptional AoE, excellent initiations, and versatilty in terms of builds. I've learned to love each of his abilities, and use them in efficient secquence to bring down the enemy. I am however not the best Amumu player out there, probably not even amongst the top 10. But with the experience and expertise I have so far displayed with the champion, I hope to be able to help you become better too.
If you have some of your own tips and tricks that you wish to share, please feel free to share in the comment section below, I am also open to all critique and feedback concerning the guide, as it is my first.
Enjoy!

Pros

Strong AoE
Scales well into late game
Very good initiations due to Bandage Toss and Curse of the Sad Mummy
Bandage Toss is also a great gap closer, for fleeing or persuing an enemy
High survivabilty
Adorable

Cons

Weak early game
Extremely dependent on blue buff
Easily counter jungled
Must be near his enemies to damage them.
Has few escape abilities (Yes, an accurate Q or ultimate could work, but the latter would be quite wasteful)

Regardless of the way you build Amumu, you want to start off in the jungle. This is primarily because Amumu is a very mana hungry champion and a poor farmer early game.
Most junglers these days start at wolves or wraiths, but when jungling as Amumu you always want to start at the blue golem. The mana regen and cooldown reduction is priceless for the little yordle, whom with blue buff can then proceed to clearing the wolf camp in a jiffy!
After wolves, he will follow the standard route - wraiths, golems, red buff, and wraiths again, making sure to pick up his Bandage Toss first at lvl 4 when he is ready to gank (see skill order).
On a side note, make sure that your team is well guarding your blue. Due to his dependency on the buff, losing it to the enemy would be DEVASTATING.

I have many different builds for Amumu, but I subconciously always tend to drift towards the AP style build, becaus let's face it - we all want to put out some damage! (I am aware of the August patch's nerfs to Amumu's AP scaling, and I understand you might be skeptical. But don't let this discourage you, the changes weren't that harsh, and they have already compensated by greatly reducing the cost of tantrum) Try out the build and you will see just how well it works!

For Items I usually go the standard Cloth Armor and Health Potion x5, although I am aware that many people go Regrowth Pendant to be able to go for a Philosopher's Stone. Personally I think this makes Amumu too squishy!

After my first gank at lvl 4 I take Boots, and depending on my success rate I will either grab some more pots and return to ganking, or start my Rod of Ages.

I try to rush for the Rod. The health, mana, and AP all come in handy. Together with this variety of stats, and the fact that it needs 10 minutes to charge up, is why I choose it first. I always build Catalyst of Aeons first, rather than the Blasting Wand, since this is great for sustain in the jungle and quite frankly the only other reason why i don't go for the Philosopher's.

Before my Rod is fully complete, I often complete my boots. I always go Boots of Swiftness as Amumu for three reasons.
1) I can gank quicker and more often
2) I can get away easy (Amumu does not have any escape abilities, unless you are prepared to blow your ulti)
3) Fleeing enemies cannot escape you! For amumu to deal damage he has to constantly be up in his enemy's face.

Rylai's Crystal Scepter, this item provides an additional amount of HP to the already quite tanky champion, which will allow you to go full AP without actually becoming "squishy". Not only will this boost your survivability and thus able you to dish out more damage in team fights, but your enemies will be slightly slowed by your AoE. Combined with your Boots of Swiftness, fleeing will be futile.

Abyssal Mask is the bread and butter for AP Amumu. It offers him further survivability, along with increased AP and the decreasing of enemy magic resistance. along with his passive, this subsitutes for great magic penetration.

Now it's time for a Rabadon's Deathcap to boost your AP right over that edge! make sure to pick this item up after you already have developed some substantial AP and survivabilty, due to the items Unique Passive, and since it offers no defensive stats.

Lastly, choose between a Will of the Ancients and a Voidstaff. If your AP mid already has WotA, go for the latter. Otherwise, Will of the Ancients is a great item, giving you the ability to stay in teamfights all the longer through the significant amount of AoE spellvamping you will do.

Warmog's Armor, Force of Nature, Banshee's Veil, and of course Sunfire Aegis are all excellent choices for our little friend, but not if you want to leave craters on the battlefield!

Smite; Essential for jungling. TAKE IT!

Flash; This ability will let you get away or prevent enemies from getting away over walls and terrain. It is also excellent combined with your ulti if you are unable to Bandage Toss (if it's on cooldown, or if you just missed it xD)

Ghost; I never pick this, but I see a lot of people doing it. I don't like running Amumu without flash, and Rylai's and Boots of Swiftness are enough to "cling" to the enemy.

This is a personal preference. I wanna have enough armor to sustain well in the jungle, while the AP per level blues help add onto my end game nuke. The flat AP quints and flat MP reds then compensate for my weak early game.

This is, like my rune page, a personal preference. I only find it neccessary to go as far as magic penetration in the offensive tree, so I do not sacrifice survivability and cooldown reduction picked up in the defensive tree.

Despair, your "W" abilty, is ideal for taking on Amumu's first target - blue golem - due to its percentage drain.
Once you have it though, it becomes more important to max your Tantrum, or your "E" ability, for faster clearing of the other camps, as well as providing nuke for your ganks.
Choose to level Tantrum again at lvl 3, because Bandage Toss is useless in actually clearing the jungle, and you will often hit the small lizards by red buff instead of the larger lizard (I've had my share of experience).
When you're out of the jungle and have ganked. Take turns by maxing both your E and your Q so that they are on par with each other. If you only max your Q you will be rather useless if you miss the ability, or once its been used. If you only max your E, the long Q coooldown will mean less mobilty and less "openings" for you to seize and intitiate. Missing a toss will be incredibly frustrating if the cooldown is high. Your Q also provides a high base damage which you will miss out on.
Like with most champions, take your ultimate, Curse of the Sad Mummy, whenever it's up.

Some important advice to jungling as Amumu is, like I stressed earlier, never to have your blue buff stolen by the enemy team. Always make sure your team has a defensive overview of the terrain around you.
Secondly, Amumu is easily counter jungled. Champions like Lee Sin, Udyr, Shyvanna and Dr. Mundo will crush you if they run into you. So YOU shouldn't try to counter jungle early game, and you should be mindful of your surroundings in your own jungle.
Lastly, Use your abilities in effective sequence to reduce jungle clearing time. Q over terrain to reach camps, and if a camp you are atttacking is low enough, begin moving away and let your W and E take care of them as you go.

PASSIVE: Amumu's autoattacks reduce the target's magic resistance by 15 / 25 / 35 for 4 seconds.

Take note that the debuff doesn't stack, but refreshes with an autoattack.




Bandage Toss(Q): (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.

Range: 1,100

Cost: 80 / 90 / 100 / 110 / 120 mana

Cooldown: 16 / 14 / 12 / 10 / 8 seconds

Magic Damage: 80 / 140 / 200 / 260 / 320 (+1.0 per ability power)

It is very important that you land this ability, as it might decide the fate of a teammate or an enemy player. When ganking, do not do the mistake of initiating with this spell unless you are ABSOLUTELY CERTAIN it will hit. Most likely it will not, and you will run shamefully back where you came from (that, or you will run like a crazy bastard and proceed with the gank). Instead, run out of the bush and use it as the enemy attempts to flee. Scare the enemy into a flight path - the skill is easiest to land when they are running away. Who knows, you might even trigger a premature flash! Make sure you don't hesitate too much because once they have covered some ground and realize, "Oh, it's you!" - they will try to juke you as they run back to tower.
In teamfights, try to initiate with this spell on the enemy AD/AP carry, to ensure they are taken down quickly.



Despair(W): (Toggle): When toggled on, Amumu will be surrounded by a small damaging pool of tears (ie *crying*). 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
Radius of AoE: 300

Base Magic Damage: 8 / 12 / 16 / 20 / 24

Max Health to Damage Ratio: 1.5 / 1.8 / 2.1 / 2.4 / 2.7% (+0.01% per ability power)

Although this abilty can drain your mana in the blink of an eye early game (and is thus the reason why you are so dependent on blue buff), always turn it on when an enemy is in range (unless you can execute them with another ability). I often end up picking up kills with this ability, especially when I don't have mana for another bandage toss, or enough range to autoattack them down. Although you may have your doubts, when end game comes you will see its true power!



Tantrum (E): (Passive): Amumu takes reduced physical damage from autoattacks and abilities.

(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: 50 mana
Radius of AoE: 200

Physical Damage Reduction: 2 / 4 / 6 / 8 / 10

Cooldown: 10 / 9 / 8 / 7 / 6 seconds

Magic Damage: 75 / 100 / 125 / 150 / 175 (+0.5 per ability power)

Unlike with most champions, there is a large benefit to getting focused by creeps, or even champions. At many point in time the abilty will become close to spammable, so make sure to seize those oppurtunities, and position yourself where you can hit a lot of enemies with it (normally this won't be an issue if the ability is spammable, seeing as it probably means you are surrounded by them).



Curse of the Sad Mummy(R): (Active): Amumu entangles surrounding enemy units, dealing magic damage and rendering them unable to move or use autoattacks for 2 seconds.

Radius of AoE: 600

Cost: 100 / 150 / 200 mana

Cooldown: 170 / 150 / 130 seconds

Magic Damage: 150 / 250 / 350 (+1.0 per ability power)

This skill has a very, very large reach, so be sure to exploit it to prevent enemies from escaping with either Ghost or Flash. In fact, if the champion you are focusing is right next to you, and they Flash, they will still be in range of this phenomenal ulti! In teamfights, make sure you pin down as many enemies as possible, prioritising their carries so that your team may swiftly finish them. Do not get overly confident however and stray away from your team. Make sure that when you bandage toss and ulti, that your team is close behind. Regardless of its huge AoE damage, you will not be able to take them all 5 down alone unless you are extremely fed, and have a few levels on them.
Your ulti can also be combined with other ultis, from other champions. Try experimenting with a friend or team, synchronizing the abilty with eg Miss Fortune's, Nunu & Willump's, Xerath's, Kennen's, or even Annie's ulti.

That is all! Email me or comment if you have any questions or constructive feedback!
See you on the battlefield!
-littalwabbit