7/19/2011:
Made various changes to the build taking advice from comments and so on.
Testing out Frozen Heart in my build now.
7/20/11:
I swapped out warmog's armor with a frozen heart for the increased armor and extra mana which is very useful.

This is a tank build. If you have no experience with tanks the you will have little to no success with this build. For the full experience of the build please have some sort of knowledge with tanks and don't try to come out of the game with the most kills in the games.

One great man once said "DUDE WTF AMUMU IS ****ING OP AS HELL" as I then replied "Lol". Amumu is in my honest opinion one of the best tanks in LoL if built correctly. I enter so many games and I find Amumus' building completely wrong and it is inexcusable. Amumu is not an off-tank, nor is he an ap nuke. Amumu is a tank. Sure you can build him as an off-tank or as an ap nuke but in my opinion an Amumu tank is the only "right" way to play him. So many times I see Amumus' building full HP or trying to go ap as a fail nuke. Sure if built correctly he can be a huge nuke and even with my guide he's a huge nuke even without any sort of damage increasing items or runes or masteries (sort of). So just keep in mind as you read that this is a pure tank build and is not meant for a solo lane whether it be mid or top with a jungler. Please enjoy!

Pros:
Very Very durable
Great ganker
One of the best ultimates in the game
Very good at soloing champs
Very good damage output
Great support
Amazing tank

Cons:
Usually focused early game
Can get shutdown from creeps easily
No taunts
Long CD ultimate
Banned ALOT from ranked games

For my marks I like to go straight magic penetration for the increase in damage on your abilities. It may be slight but it will all be very noticeable when in the team fight your despair will be able to just melt straight through their tanks magic resist and do some massive, massive amounts of damage. As for seals I like to go for armor just for the pure survivability early game as well as late game. Also for the obvious reason that you are a tank so obviously armor would help out a ton. For glyphs I like to go into magic resist at level 18. Though this is semi situational it helps nonetheless especially against casters and mages. For qunits I go straight up health. This adds 76 more health at the beginning of the game which is tremendous. This will get you almost 900 hp in the very beginning of the game which is as far as I know the highest amount of any champ.

For masteries I go the typical 0/21/9 build for maximum defense as well as I pick up the extra experience in the utility tree. The defense can be changed for whatever kind of team you are facing. I like the extra experience in the utility tree so I can keep up in level with some of the other ap casters on their team. As for summoner spells I like to go with ghost and clarity. These are both very useful spells for Amumu both early game and late game. Your ghost allows you to chase people down, catch up to teammates, setup for ganks, and escape. Clarity is just all around useful as you do occasionally go OOM endgame and very often go OOM early and mid game. This is due to the fact that despair is most likely going to be on most of the time when in a conflict as well as mashing the "E" key for a tantrum every time off cooldown. This means that if you're in a team fight that lasts lets say 13 seconds then the mana you would on average use is:
(80) + (8*13) + (50*(roughly)4) + (200)
The mana cost for the average team fight is roughly 584 mana. This is obviously an approximate amount because there is no way I could possibly calculate how many people are hitting you (reduces the CD on tantrum) and how fast they are hitting you. Also you may or may not use your ultimate but you usually would end up using it for the sheer Crowd control it brings to the fight. By this point you will be well under half of your mana and turning despair off is not really worth the time and effort. This is why I choose Clarity over things like Exhaust and Ignite. Leaves those things to your carries. You can also turn this into a full out jungle build using the same build by just swapping out Ghost with Smite. It's all basically the same as laning but beginning game you are in the jungle.
EDIT!
Instead of Clarity I now take Flash it has proved to be more useful and all around better.

For items I usually like to get a Ruby Crystal for a nice boost in health in beginning game. This can be swapped out though with something else like perhaps Boots if you really want to but I don't find boots this early too great for Amumu. Your fist base back I'd say should be around 1000 gold or later so you can pick up your Mobility Boots and be the next speedy Gonzalez. With these boots out of combat you will run a cool 435 movement speed which far exceeds what you would with Boot of Swiftness which would bring you to about 400 or 410 I'm not sure but it will still be in effect when in combat which Mobility Boots will not, but that's why I like ghost because if you really need to chase someone down then you have Ghost as well as Bandage Toss which will give you a stun and pull you towards them. I then base when i hit around 1400 to get a Giant's Belt for a lot more hp and to also help me build up for an amazing item, the Sunfire Aegis. This along with your passive Cursed Touch will be able to do massive amounts of damage. I then continue to farm minions as well as get assists and kills when possible until i have enough for either another Giant's Belt or the Complete Warmog's Armor which is around 2500 gold with you already having the Ruby Crystal. Once you obtain this armor you're going to want to farm a lot of minions to increase its permanent passive. After you get Warmog's Armor the items will differentiate depending on who is in your game from here on. If there is an AP carry on the opposing team like Mordekaiser, Brand(this guy is a ***** to deal with), LeBlanc, then I'd suggest getting a Force of Nature earlier than your Randuin's Omen for the large increase in magic resist. With that being said let's say your facing a team such as this one: Vayne, Warwick, Janna, Rammus, Kassadin, you would probably want to get a Thornmail first then your Randuin's Omen for the maximum protection from their carry while doing a lot of damage through Thornmail. You would then want to get your Force of Nature to finish off the build which will give you protection from that Kassadin and Janna. Sometimes you may be facing a team with no AP characters at all. Once you get your Thornmail then you would probably buy your Force of Nature next. But if their team has no AP then it would become useless, you should then swap out the Force of Nature with either another Warmog's Armor or a Banshee's Veil.

NOTE: I am now using a Frozen Heart in my build instead of the Warmog's Armor for the extra armor and mana. I am leaving the information above the same as a warmog's will still be viable it is just personal preference. I am still testing out other items as well.

Right off the bat I'd say always get Tantrum, even though Despair may look very appeasing for farming creeps you will pick that up at level 2 anyways. At level 3 you should pick up Bandage Toss so that you may now have a reliable stun and gank tactic. This is also a great initiator for a good ol' fashioned brawl because not only does it pull you towards the target hit, it will also stun for one second. I then continue to level up my tantrum whenever possible for the decrease in physical damage taken. Afterwards focus on getting Despair to rank 5. You should always level up Curse of the Sad Mummy whenever you get the chance(lvls.6/11/16). Afterwards focus on your Bandage Toss until level 18.

You are definitely better off laning with some people more than others. As an Amumu you definitely would never want to lane with another tank. This is pretty much a given but nowadays it seems like whenever I enter a game no one gives a damn. Amumu is very effectively used with a ranged carry such as people like Corki, Vayne, Caitlyn, and Ashe. Amumu is also amazing with a glass cannon type ap carry like Swain, Brand, Anivia. Below is a list of champions that go great with Amumu and who completely suck *** with Amumu.(Not all characters will be named use your better judgement for any unnamed champs)


Ideal Partners while laning:
(AP)

Don't do it:


As you may have noticed most of the bad laning partners are both melee and tanky or in vlad's case ranged and tanky. Regardless another tanky character with Amumu will never end out positive because the enemy will be so confused about who to attack that they will most likely just play it safe because they don't feel they can do enough damage. I personally like to lane with the more Nuke-ish type characters because the amount of CC Amumu brings with him by the time he hits level 6, is unprecedented by any other champion in the game in my opinion. With a good brand all it takes is one good bandage toss and he can take care of the rest you just have to take the hits for him. My personal favorite "laning parter" (I use quotes there because he is not actually my laning partner but it is amazing) is Trundle. The jungle trundle is probably one of the best junglers other than Lee Sin for a gank in a lane with an Amumu. All it takes is some cooperation and you just got your team a double kill and they just fed the troll. I do think that Trundle is one of the best junglers to have on your team with you if you are indeed playing a non jungle Amumu, but that of course is my opinion.

NEVER EVER USE YOUR ULTIMATE IN A 1V1 FIGHT... EVER. Now that that is out of the way, Amumu's ult brings the most CC to the fight in one spell in the game. It snares, stuns, damages, pacifies, and silences everyone caught within the area of the spell... which is huge. Back to the 1v1 situation now, your despair and tantrum should be able to do enough damage to either hold the enemy off, kill them or give you enough time for reinforcements. Amumu is a great initiator for team fights. Notice I said "TEAM FIGHTS" not fights in general, leave that **** to Xin Zhao. To properly initiate a TEAM fight which requires MORE than 2 people on both sides you need to land a Bandage Toss usually on either their AD carry, AP carry, or as a last resort support. You always want to try to hit a carry first as it stuns them then your despair is melting away at their health, then the rest of your team jumps on the one you bandage tossed. Note: If your team is somehow dumb enough to attack the tank of the other team please feel free to not use you ult in the fight because you will lose anyways. If you are not focused in the fight which you most likely will be than you should quickly either use your ultimate to try and grab their aggro or target the carry and start blowing them up. This is because you don't want your carries and support getting blown up. And if they aren't focusing on them then they are focusing on you or your jungler, which is just fine. No jungler in the game is squishy so it honestly would be better off he get attacked than your carries. This even includes Fiddlesticks because of his drain making him nigh unkillable. When you hopefully win said team fight there will usually be someone running or they already ran. If they already ran and it's later in the game then it's basically a free Baron, but, if they only ran once they lost the fight, they won't be living too much longer with a well placed bandage toss. This will pull you towards them, stun them and either allow your team to catch up to them or allow you to run along side them scoring you the ace.

Amumu, the nightmare of all carries, the bringer of CC and team fight victories. These are the reason why Amumu is banned almost every ranked game. He has alot of CC and will almost always be the reason a team fight will go in your favor. In a ranked game it is very rare to see a lane Amumu most of the time there will be a jungle Amumu which I end up using almost this exact build for anyways because then you knock out 2 roles with one pick; tank and jungle. All you need to do is grab Smite over Ghost and start at blue. If you do end up laning with Amumu which is by no means a disadvantage at all, then continue with your regular routine of last hitting and harassing. When laning in bottom lane you are usually with a carry, when this happens I usually let them farm up whilst I get some last hits every now and then. This is because the more fed your carry is the better your team will be late game and this build honestly doesn't require too much farm early game, and late game you'll be farming like a pro with despair, tantrum, and sunfire cape. And also be sure to always help your jungle pick up dragon on every respawn.

DO NOT EVER GET NINJA TABI! This reduces the cooldown effect of your Tantrum ability which sucks. DON'T DO IT!
If you can get a clear shot on a carry with a bandage toss then take it even if you miss with it you will cause them to become very paranoid and most likely back off for a little while.
NEVER USE YOU ULTIMATE ON ONE PERSON EVEN IF YOU ARE CHASING SOMEONE DOWN FOR A GANK AND YOU WILL GET THE KILL IT WON'T BE WORTH IT FOR THE DOUBLE LANE DOUBLE KILL!
Please Leave your comments and feedback on this guide and don't comment on the guide if you had not used it.