1. Changed Masteries, Rewrote masteries chapter (thanks to bajakomza)
2. Changed a set of runes and added situational item chapter (thanks to Mastajdog)
3. Changed Masteries yet again for the patch
4. Rewrote Masteries chapter
5. Added Pros/Cons chapter
6. Added a build that was more inclined to AP
7. Added a whole new series of chapters talking about the AP build (it's at the very end)
8. Changed the name of the "Changes" chapter to "Change Log"
9. Added a few more items into the "Situational" chapter
10. Changed the Item Purchases for the AP build so that it shows the order that I buy the items in

I am writing this because I find Amumu generally fun to play with because I'm that kind of person who likes playing with mummy midgets. Please comment on how to improve this guide and thank you for taking your time to read this guide. AND to reinforce your memories, PLEASE VOTE AND COMMENT, because now, I really don't know what else to add to the guide.

Pros:

Can Farm really well

Ult is great for team fights

Can chase really well with your Q (Bandage Toss)

Can escape ganks with Ult

Has an Aoe Madreds Bloodrazor (His W)

Great AP ratios except for his W

EPIC ARMOR!!!

Cons:

Will be called OP, many games that I have played with Amumu had responses from the Enemy like this: "OMG, AMUMU SO OP, AOE MADREDS??? WTF???" "OMG, AMUMU SO OP, HAS DOUBLE STUN!!!"

Screwed when you run out of mana

Pretty much no chance of playing him in Ranked because he will always be banned

Bandage Toss is easy to Juke

Yeah, I'm biased.

The items you should be getting first are Glacial Shroud, Mercury's Treads and an early Giant's Belt for building up into your Warmog's Armor.
The reason for this is because:

Glacial Shroud because it gives you lovely armor, CDR and 400+ extra mana. You will then build this into Frozen Heart late game, but this item gives you the ability to spam your Tantrum and keeps you rather healthy in early skirmishes.

Mercury's Treads give you magic resist to those annoying mages, tenacity to get out of CCs early and gives you speed. Without this, you will be really easy to pick off.

Giant's Belt gives you quite a lot of health and gets you closer to your off-tank potential. Also, you would need it to build into Warmog's Armor later.


After that, you can then purchase a Warmog's Armor giving you a LOT of health. Then you would get another Giant's Belt to build up to Rylai's Crystal Scepter. Once you get Rylai's Crystal Scepter then you would be pretty damn pro, slowing enemy champs with your Despair and Tantrum constantly. Then you get Lich Bane, bringing your percentage of Despair to 1 extra percent. Also, because Tantrum is spammable, the Unique passive for Lich Bane will be giving you 200+ extra damage with your hits. The Atma's Impaler is very random but actually works. Because of your insane health, this gives you 81 extra attack damage which lets you devastate them with your auto-attacks as well as spells. Your last item should be Frozen Heart giving you a great deal of CDR, Armor and mana.

Thanks to Mastajdog yet again, I have set up a Situational Items chapter. Feel free to comment and suggest some of your own.
The situational items are:
Banshee's Veil because it gives good health, mana, a lot of MR and a lovely Unique Passive which blocks a negative spell every 45 seconds. I would only suggest taking it, if there is a nuker out there always focusing on you, or the opposing team has lots of mages.

Force of Nature because it gives very decent health regen, great MR, extra speed and an insane Unique Passive. Again like Banshee's Veil this is a situational item.

Abyssal Mask because of the AP it gives with the MR. Great for when you need the extra MR but don't want to lose AP or you want to elevate it.

Sunfire Aegis because it gives you good health, good armor and the Unique Passive works well with your Despair.

Zhonya's Hourglass because of the exceptional AP, good armor and great Unique Active that lets you laugh at them while they try to take that "OP-DOUBLE STUN-AMUMU" out when you initiate.

Put a lot into health and a lot into ability power per level because Amumu is supposed to be an off-tank dps.

I get points into Armor because I really want to laugh into my opponents face and force them to get some Armor Penetration. I put points into Health because it benefits my Atma's Impaler. Putting points into Indomitable and Tough Skin creates synergy with Tantrum reducing damage taken from minions to 14 and damage taken from champions to 12. Getting points into Honor Guard further amplifies this effect. Points into ability power and MR penetration don't need to be explained. Points into Sorcery and Enlightment gives you massive Cooldown Reduction.

Well, this really depends on how your match is like, but normally I would do my skills in this order. Max Tantrum first because the passive is great and is great for farming. Maxing Despair 2nd should make taking champs down a tad bit easier. Getting a point in Bandage Toss first helps you initiate and is a great CC. You should be taking points into your Ultimate whenever you can.

Teleport is for after you died and you need to get into lane quickly because it is vulnerable. It can also be used for flashing into other lanes when you need to. Ghost is for chasing people and keeping them in the range of your Despair longer. This is only used for this purpose till you get Rylai's Crystal Scepter. Also this spell is great for running away from ganks. Say for example that you are ganked by 4 champions at once. You can just Ult then ghost to run away. It might be a bit of a waste for you Ult but it prevented you from dying so it was worth it.

Early Game:
During early game, you should be farming, farming and farming. When you see an enemy champion, do not engage unless you know you can secure the kill.

Mid Game:
During Mid Game, you should be building up into your Rylai's Crystal Scepter. You should have been gaining massive amounts of gold because of your Tantrum.

Late Game:
During Late Game, you should have all your items, so you should be very tanky while doing a hell load of physical and magical damage. If you can't take on an enemy champion now, then you're doing something wrong.

In huge Team-Fights, you should be using your Bandage Toss to initiate and then using your Ultimate, attacking someone for the Lich Bane Passive, Despair and then attacking again and Tantrum with another auto-attack. With your full build, you should be doing over 1500 damage in a very short period of time. Remember to Ult the enemy team when you are sure, you can land Despair and Tantrum on them or it would be rather wasted.
In 1v1s you should Bandage Toss in and activating Despair at the same time. After you attacked him once then Tantrum and keep on attacking him. If he tries escaping, then you can ghost for extra speed to keep him within the range of your Despair even though you shouldn't be doing this because he is constantly slowed by your Despair.

I think that with this build Amumu would be really good because you have lots of health, Magic Resist and Armor, while dealing lots of damage due to Atma's Impaler and Lich Bane, and of course you should be burning down that health bar with your skills. Thank You again for spending your time reading this. Also, as a final note, PLEASE vote and comment. You can also request for a guide on specific champions that you want.

I know what this looks like, you're probably thinking: WHAT A CHAPTER AFTER SUMMARY, DID HE PLACE IT INCORRECTLY??? So to answer that question: No. I fully intended to put this here because it doesn't actually fit with anything above. I was experimenting with some builds and found that this one actually works great if you are willing to give up the EPIC ARMOR and 4000 health. So, to start this off, I will tell you exactly how much AP you will be getting from this build, the benefits of this build etc.

I looked at this and found it pretty decent.

Health: 3114

AP (because the stats for the AP up there are screwed up): 670 (not including masteries)

Mana: 3015

How much your abilities do at Level 18:

Bandage Toss= 990 raw damage +1 sec stun

Despair=5.7% of max health per sec dealt in damage + 24 raw damage

Tantrum= 510 raw damage

Curse of the Sad Mummy= 1020 raw damage + 2 sec Aoe stun

Archangel's Staff gives a lot of AP and mana due to its AWESOME passive. You should rush these to avoid running out of mana early game.
Ionian Boots of Lucidity is great because of the extra speed and the CDR it gives you.
Lich Bane, this is what makes you so scary. I'll give you a scenario, you have full build, you are Level 18. You are roaming around when you see an enemy champion. This is what happens, you Bandage Toss over to him and you hit him. That Auto-Attack just did 877 additional damage... Yes, the Unique Passive is downright awesome. You get adequate mana, good AP and an AP Sheen, this is a must on AP Amumu.
Rod of Ages is something that needs to be rushed. It should be your second item after your boots because you get the benefits, the longer you have it. It gives a LOT of mana, a LOT, of health and good AP. Plus the Unique Passive helps your lane sustain.
Rylai's Crystal Scepter, this should be your 4th item. I can't stress how good this is on Amumu, it is what lets you slow the enemy constantly with your Despair and Tantrum. It has a lovely Unique Passive, great AP and lots of health.
And of course, the last but not least, Rabadon's Deathcap. AP and Unique Passive. Enough said.

I will talk about items that are good as well:

Abyssal Mask, this is for when you need the extra MR but you don't want to lose that much AP.
Sorcerer's Shoes, people say that this is better on Amumu than Ionian Boots of Lucidity and it is very useful because of the extra MR Penatration.
Zhonya's Hourglass is a good buy when the enemy is full of AD carries and they get far too annoying for your liking. Unique Active and Armor help a lot. Also, the huge amount of AP is great.

Great for people who just want to get in there and destroy everyone they meet. It is very scary when fully built up and can literally earn you easy pentakills. Last words? TANKED NUKER!!!