This is my Amumu build. *dramatic music*
Please don't comment if you don't have anything nice to say, I create these as reminders to myself and to help out my friends. You don't need to like it, but if you do I would love to hear your feedback. :)

I use the same combination of runes for all my Ability Power characters so it may not be the best for particular champions, but it covers the basics and works to my play-style.

Amumu is a tank. Ergo, he needs his points in the Defense tree. Some people build using the 21-9 build, I prefer to pump all my points into the Defense tree to make Amumu able to take as much damage as possible.
I ignored Initiator and Mercenary because they do nothing towards beefing up Amumu. Initiator doesn't help because I give Amumu Movement Speed 3 boots anyway so that covers chasing and Mercenary is just for gold gain, which if you play a good Amumu you will never need.

The idea behind my build is health health and more health, with plenty of regen to keep it as close to full as possible. An Amumu who can just soak up any kind of damage into his seemingly endless supply of health is scary. Because he can sit right next to you attacking you while his Despair and later Sunfire Cap are hammering away at your health. And Amumu gets two stuns as well as this build gives you movement speed 3, so he is very hard to run away from once you realize you are losing.
Early Cloth Armor is good for Jungling and early lane sustainability because you get to buy 5 Health Potions at the start (if you so wish). You also eventually turn it into Thornmail, so it doesn't go to waste.
The reason for Philosopher's Stone, which I turn into Eleisea's Miracle is for that essential Health Regen and Mana Regen because keeping Despair active burns through your Mana.
Warmog's Armor, although expensive, is a fantastic first item to get fully built for Amumu. It gives him a stack of health early, health regen and it builds up further so the earlier you get it the earlier you get the full bonus.
Boots of Swiftness are great for Amumu, it helps him outpace most champions and keep pace with almost all the rest (Excluding Janna and people who spam Phantom Dancer). Keeping up with Champions is especially important for Amumu because his main source of damage dealing will be his Despair whittling away at all surrounding enemy's health.
With the first three items built being aimed at keeping Amumu in lane and effective now it's time to build items to improve upon what is already there. Sunfire Cap improves Amumu's already considerable health and gives him a bit of Armor for those annoying Attack Damage champions. It also works perfectly in conjunction with his Despair ability, creating a powerful aura of damage around Amumu.
Thornmail assures that no Attack Damage champion has a hope of soloing you, which is a considerable feat if you take into account I even mean a well fed Tryndamere or Yi. (I used this build in a game where I ended up pushing a lane against a Tryndamere and Jax and didn't even need to directly attack them, my items and Despair did all the work for me.)
Finally, if you get around to building it, Force of Nature gives you a tonne of extra Health Regen, which you might need in a game that has gone on long enough for you to be building it. The extra Magic Resist is useful too, although I've not found any Ability Power champions who Amumu struggles with after building Warmog's Armor.

Coming Soon!

Coming Soon!

I hope anyone who has read this enjoys the build and makes use of it to positively influence their Amumu game. He is the best tank in the game in my opinion, a powerful force alone, devastating in any team fight, and utterly unstoppable when fed.
Within the next couple of days I will update this with the 'Skill Order' and 'Some Useful Tips' sections filled out.

~Saamun.