If you don't like my build please leave constructive criticism so I can improve. Vote up if you liked the guide. Please don't vote down without trying it first.

I wrote this guide after being inspired by Juffalo's.
I used his build for at least a week and then began to improve on it. All my knowledge is accredited to him and in-game experience.

Amumu: The Tank Slayer
I find that this name is pretty self-explanatory. The enemy thinks they are going to have a good tank with that overly-fed Cho'Gath, NOPE. With this build you don't care about enemies with armour and magic resist because you go straight to the health, % damage is the best thing about Amumu. He is viable to almost all pre-made teams as he is an amazing initiator, does amazing damage over time, and he refuses to go down if played correctly. I play Amumu as an off-tank dps.

I understand that this build may not be perfect or the best build for everyone but this is what works for me and I believe that if you follow this guide properly you will be able to pull of a pretty good Amumu.

Edit: August 10th, 2011: 1 million views, thanks everyone, it really means a lot

For Amumu I only use the top two Summoner skills but there are others that are viable.

Teleport: Very useful in farming and very useful for ganking if you have competent teammates. If your team has good map control you will have sight wards covering most of the key points on the map and you will be able to pop up behind almost anyone unseen. You can also use this to Teleport to large minion waves for farming and to turrets in case they are being back-doored. Knowing when to Teleport and when to walk is very important, you shouldn't always waste it just because you have it up.

Flash is the second Summoner Skill I use which helps me to aim my ultimate much better which can make or break a team fight. I also use Flash as an escape mechanism in case I'm being ganked, or I just can't go 1v1 against whoever I'm fighting. I also use this to get past minions to get an unsuspecting champion with my Bandage Toss without him being able to dodge it. Many people don't understand that using Flash with Amumu is not because you need a crutch; rather it allows you to go massively more aggressive.

Ghost is a very good Summoner Skill for Amumu, if Flash is removed eventually I will be switching to Ghost.

Exhaust is the last Summoner Spell I’d recommend, I see many Amumus take it and it does help a bit since the removal of Rylai's Crystal Scepter but I still don’t take it.

I always start off with a Regrowth Pendant and a Health Potion or Mana Potion. Once I get enough gold I recall and grab my Philosopher's Stone or I stay in the lane until I can afford my boots, depending on how the lane is going. I always teleport back into the lane right after buying or to another lane if it needs help. Those are the only things that are important about the item buying that I can describe, other than those tips follow the build or branch off depending on their composition.

Optional Items to build:
If you are facing heavy Attack damage than you will want to build Thornmail, Frozen Heart, Randuin's Omen, etc. Any item after Eleisa's Miracle can be substituted.

If you are facing heavy Magic damage you want to build a Force of Nature or a Quicksilver Sash if they have heavy CC. Any item after Eleisa's Miracle can be substituted.

Also, if you're facing a team with very little CC, you don't need Eleisa's Miracle so you can upgrade your Philosopher's Stone into Shurelya's Battlesong for the Cooldown reduction and the helpful active. Thanks CaioYon.

Bandage Toss: Skill-shot, aim, click. You want to lead this a little bit and if you aren't close to a wall it won’t pull you through even if you hit your target. I often use this after my teammate that I lane with initiates so he can unload a full combo. You can also drop unexpected squishes with those unexpected impossibly far skill-shots followed by a Tantrum. Don't be upset if you miss your Bandage Toss when you’re first starting out as Amumu], as it is skill-shot that travels differently than some other skillshots. You’ll notice a difficulty in hitting with it if you’re new to Amumu and if you play champions like Dr. Mundo or Ezreal, whose skill shots move very quickly.

Despair: Amumu cries on his enemies doing percent damage per second + magic damage. Stacks very well with your Sunfire Cape and becomes even stronger once you get your Abyssal Mask because of their reduced Magic Resist.

Tantrum: Activate to break dance and deal damage. Getting hit reduces the cool down. Passively reduces damage taken. Great farming skill, huge source of AoE damage. Stacks very well From Juffalo

Curse of the Sad Mummy: This is an amazing skill. When activated it will do a massive AoE snare for 2 seconds, the enemies take damage instantly and are completely unable to move for the ultimate's duration. Don't be afraid to use this skill, after a while you will know if it's a good time to activate it to keep them from running, or to let the lucky Master Yi run back to base with 14 health. This skill is nice to use in combination with Flash as you can aim where to drop your ultimate much better or push yourself a tiny bit farther into position to hit all 5 champions, rather than only 3.

Early-game:

Often I find myself in the scenario where someone has been harassing for a while early game and I decide to NOPE them harder than that one time they really wanted the cool action figure with Kung-Fu grip. I would Bandage Toss, cry my life off, throw a Tantrum, curse them, and if I timed it correctly I should be able to stun them again before they are able to run away leaving them with another Tantrum and a few more seconds of Despair. Depending on my laning partner they will be either super dead, really dead, or at 25 health. If they are extremely low health and you are still almost full you can go in behind their turret, Bandage Toss (don't miss) and either 1-hit them, or Tantrum and flash out of turret range.
Although this seems extremely unlikely it's really easy to pull off against morons that overextend, most of the time it's a good thing to let them push you to turret, just don't let them do significant damage to it. *Level 6 minimum*

Also, when you are facing heavy melee auto-attackers that are overextending, I.e. Master Yi, Tryndamere, Jax, you are able to Bandage Toss and then Tantrum and run back to safety without being damaged. This is a good way to harass with Amumu at a low level without taking much damage. At this time you should have already ported out and back in for your Philosopher’s Stone and should have the Health Regen and Mana Regen to heal if you got damaged a bit in the process or get damaged when using Tantrum to farm.

The entire goal for the early game as Amumu is to farm as much gold as possible and try to help your laning partner get as many kills as possible.

Mid-game:

During the Mid-game you want to focus on getting the Golem Buff as much as possible, ganking as many people as possible, and pushing as many turrets as possible. Team fight are a must in the Mid-game and you are the key to winning those. If you can time your Curse of the Sad Mummy properly your team should be able to take out at least 3 enemies in a team-fight. If things go bad you are pretty good at escaping due to your stun, ability to pull through walls, and do heavy damage to anyone dumb enough to follow you extremely closely.

End-game:

Delicious End-game, as Amumu you are pretty much the main target now, but if you are far enough into your build you should be strong enough to take a massive chunk out of turrets and enemies. Keep on pushing and keep on being a key part of the team fights is the key to victory. Amumu should be able to 1v1 any champion on the enemy team and by end-game can duo Baron Nashor with any other champion in the game. You are Amumu, The Tank Slayer, wear the name proud and don't let anyone on the enemy team push you around.

Marks: For Marks I use Greater Mark of Magic Penetration for the flat Magic Penetration. I find that it helps my overall damage I recommend it for Amumu. I know no alternatives, if someone would like to share, that would be nice but seeing as every Summoner should have ArPen and MPen marks, MPen is recommended.

Seals: For Seals I use Greater Seal of Scaling Mana Regeneration for the Mp5. I find that it greatly helps me stay in the lane and is very helpful whenever I'm disallowed the Golem buff.

Glyphs: For Glyphs I use Greater Glyph of Scaling Cooldown Reduction for the cooldown reduction. It helps me initiate fights more often, harass better, and farm better. This also applied to my Summoner Skills which allow me even more farming and more protection against people back-dooring turrets.

Quintessences: For Quintessences I use Greater Quintessence of Scaling Cooldown Reduction for the CD reduction. It helps me initiate fights more often, harass better, and farm better. This also applied to my summoner skills which allow me even more farming and more protection against people back-dooring turrets. I also used Greater Quintessence of Health. It really just matters on the match, and if I remember to change between the two, but whichever is on my page works for me and many people have Greater Quintessence of Health as their default rune, which is suffice to the proper workings of Amumu.

Mastery points are to be done purely how you prefer. Viable sets would be:
9/21/0
9/0/21
0/21/9
0/9/21

Summoner Skills:

Teleport: Good for farming and protecting from back-doors.
Flash: Good for running, chasing, initiating a fight, or aiming your ultimate.
Ghost: Good for chasing, but not recommended.
Exhaust: Good for chasing and is actually viable since I removed Rylai's Crystal Scepter but personally I still do not use it.

Items:
Philosopher's Stone
Boots of Swiftness
Sunfire Aegis
Eleisa's Miracle
Abyssal Mask
Guardian Angel
Banshee's Veil

Game Phases:
Early Game: First-blood, farm, farm, farm.
Mid Game: Farm, gank, team fights, push, level up.
End Game: Gank, team fights, Baron Nashor, back-dooring, Nexus, victory.

Runes:
Greater Mark of Magic Penetration
Greater Seal of Scaling Mana Regeneration
Greater Glyph of Scaling Cooldown Reduction
Greater Quintessence of Scaling Cooldown Reduction OR Greater Quintessence of Health

Pros:

Amazing initiator
Amazing farming
Can tank and have good DPS at the same time

Cons:

Squishy Early-game
People will often tell you that you are bad early game
Many kill-steals, accidental or not.
Not recommended for level 30 play

I want to thank Juffalo for writing "Amumu: The Bi-Polar Fireball" in which my guide is based on. I would have never been a successful Amumu or even been inspired to try out this sad little mummy without first reading his guide.

Thank you for reading my guide and remember if you liked it vote up and please leave your criticism if you voted down. Please don't vote down without trying this build first.

Feel free to add me if you want to play some 5v5 with a sillyguy Amumu. My Summoner name is "HttP Magick"

Also, if you're thinking about starting fresh, maybe you could use my referral link? :D

Disclaimer: I play Amumu this way and it is the best way for me, I would like to reiterate that I do not know if this is the best Amumu guide and that I greatly appreciate anyone and everyone that has helped me learn how to play Amumu.