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"ZA WARUDO! Time has stopped." Amumu is the tank that can, for 2 seconds, STOP TIME (with the exception of channeled abilities). This gives him one of the most OP ults in the game, and makes him one of the best tanks.

I play Amumu as a tank that can dish out some damage. If you like being in the center of the action, literally, you'll love Amumu. I try getting items that make my champion tankier while providing some offensive purpose. I play Amumu primarily as a jungler.

Jungling/Playstyle videos coming soon!

Pros:
ZA WARUDO! Curse of the Sad Mummy, Amumu's ulti, allows him to STOP TIME, for two seconds, everywhere within the ulti radius, while allowing you and your team to move around and MUDA MUDA MUDA MUDA! the enemy team.
Tons of damage for a tanky champion.
AOE much? Despair, Tantrum, and Curse of the Sad Mummy are all AOE spells
Assist much? AOE spells means you'll be damaging multiple champions means if any of them die you'll get the assist.
CC much? In addition to ZA WARUDO, Amumu has Bandage Toss, a 1 second stun. Combine the 2 on 1 enemy champion for a 3 second disable.
Cursed Touch: 35+ Magic Penetration when you auto-attack someone
Can somewhat carry if needed, due to his decent damage output and survivability

Cons:
Gets banned every once in a while in ranked matches
Very susceptible to counter jungling
As with most "tank" champs (Shen, Rammus, Amumu, Malphite) your usefulness is primarily dependent on your tankiness to the enemy team. If the enemy team becomes fed and/or you're not, you'll become absolutely useless in teamfights because your effectiveness comes from being able to take damage and be in the center of the enemy team causing disruption and being a ****. If this ever happens to you, farm, don't feed, and get some assists from well planned ganks.

Pros:
Tanky enough most of the game
40% Max CDR! You can spam your abilities even more!
Amumu's offensive ability is pretty great by itself, but I enhance it with CDR and Defensive items that also provide some form of offense.
Pretty good Mana Regen late game because of Runes.
Rylai's Crystal Scepter gives infinite slow with 'W'.

Cons:
Expensive, and not the tankiest build.

Most builds are too tanky, therefore only utilizing Amumu's CC and not his full offensive capabilities. Or, the builds are too squishy, where you can't survive in the middle of the teamfight and you have to play carefully from the outskirts, therefore not fully utilizing the AOE aspect of Amumu's skills. My build provides a good, strong balance between the two.

Smite: Amumu needs Smite for early game jungling. Also, Amumu is pretty good at stealing A WARDED Dragon/Baron due to his ultimate giving him 2 seconds to successfully steal. It can be used defensively or offensively in the jungle for last-hitting neutral buffs.

Flash: A great spell for every champion. Can be used to run away or chase someone running away. Also great for positioning yourself for Curse of the Sad Mummy, but if possible, just Bandage Tossing and using the ulti is much better. That way you can save Flash for escaping death or something else.

I go down the Defense Mastery and get Strength of the Ages. The rest into the Utility Tree. As for Runes, I get Armor Quintessences instead of Armor Seals, Scaling Mana Regen Seals, and Flat/Scaling CDR Glyphs.

Frozen Heart: 90 Armor, 20% CDR, and 20% reduced enemy champions' attack speed. Even if their team is primarily AP, the Armor and the 20% CDR make it a keeper in my books (not to mention the 90 Armor helps magnitudes when turret diving or tanking turrets).

Spirit Visage: The Health Regen is a little wasted on Amumu, since he isn't Garen or Dr. Mundo. However, it has the highest Magic Resistance for a single item, additionally gives Health, and the 10% CDR tops off your CDR at 40%.

Shoes:
Mercury's Treads: CC can be very annoying, preventing you from disabling an enemy champion, saving an allied champion on time, or using your ulti in key moments. Mercury's Treads won't completely eliminate CC, but it will definitely help.
Ninja Tabi: When the team is primarily AD, and they don't have a lot of CC, you should get these, and it will help you throughout the match.
Boots of Swiftness: When the enemy team has a lot of slows, you gain more benefit from an overall faster movement speed as well as the passive of 25% reduced effect of slows.
Sorceror's Boots: If your team is doing well, and you don't need to worry about dying in teamfights, this will help your damage output tremendously.

Sunfire Aegis: Synergizes well with Despair. You upgrade your base magic damage from 25Dps to 60Dps (at lvl 13 onwards). A pretty tanky item with good DPS.

Thornmail: if the enemy team has a really fed Tryndamere, Caitlyn, etc. AD Champions seem to hurt a lot more than the old days, so this item has become a part of my usual end-game build path.

Banshee's Veil: if the enemy team has a LOT of casters and CC. I'm not the greatest fan of this item because it offers no damage output of its own, but if worst comes to worst, get it. It's also slightly tankier than Abyssal Mask
Rod of Ages: HP, MP, AP.

Default ganking combo (use this, primarily, to initiate ganks early/mid game):
Bandage Toss enemy champion.
Autoattack one time for the to lower enemy champion's Magic Resistance.
Turn on Despair.
Tantrum.
Curse of the Sad Mummy.
If the enemy is pushed all the way towards your ally turret, you can just walk up to him or her and save your Bandage Toss until after they Flash or use their runaway abilities.

Protip: Don't immediately use Curse of the Sad Mummy after you Bandage Toss. You want to maximize the time that the enemy champion is disabled, and you want him/her to simmer in your Despair for as long as possible. That's why you should throw your Tantrum first, then use the Curse.

Alternative (sometimes your Bandage Toss is on cooldown, or you just really need to disable your enemy ASAP if you might miss the skillshot, or if minions are in the way, etc.):
Curse of the Sad Mummy.
ZA WARUDO! Time has stopped. Proceed to walk towards enemy champion with Despair.
Autoattack.
Turn on Despair.
Tantrum.
Bandage Toss.
Tantrum.

Keep chasing and throwing Tantrums and Bandage Tosses until he or she dies.

Protip: Always autoattack the big one first so that it gets the magic resistance debuff.

I do a pretty standard/safe path. Depending on which ones are bottom I'll Smite Krugs, go Red, Smite Blue, then go Wolves if I don't see a good gank. Or Smite Gromp, go Blue, Smite Red, then go Raptors if I don't see a good gank. This will put you at LVL 3, when you should have [[Bandage Toss] and be ready for some sick ganks.

Successful ganks with Amumu before LVL 6 are tough because his DPS is not so great, and his only hard CC is a 1 second stun. If you do gank before LVL 6, or while your ulti is on cooldown, just make sure you have the Red buff for the extra slow/damage. Once you get to LVL 6, ganking becomes much easier.

Smite the Raptor for the True Sight so you can spot out wards and gank for teammates undetected.

EARLY GAME

Just keep farming. Listen for pings. Look at your allied champions' health, and the enemy champions' health. Wait for a good gank, or defend for your allies when one of them asks you for it.

If your Ally top laner is overextended without wards, you should go through his lane into the brushes. He's asking for a gank, so he's very likely to get one. Then you can be there to counter-gank.

MID GAME
Ward jungle entrances, and whatever else feels important to have vision of at the time. Kill the scuttle crab whenever he's scuttling around to get some free vision and the movement speed buff.

Teamfights, you should be initiating, but try to initiate so that your likelihood of dying is as low as it ever will be in that situation. Also, note that you don't have to immediately use Curse of the Sad Mummy. Use your ult when your team is ready. If you ult and none of your allies do anything, then you just wasted it. Also, make sure you hit as many enemy champions as you can with your AOE spells so that you get tons of assists. Try to focus on the squishiest one and/or the one that is dealing the most damage. Try to get minion agro if you can. Minion agro = lower Tantrum cooldown. In teamfights, I basically button mash 'E', use 'R' when the moment's right, use 'Q' on someone whenever it's off cooldown, and leave 'W' turned on. If you're right in the middle of their team, you'll probably be almost dead at some point. When that happens, just use Flash towards your team, and if you're lucky, an enemy champion or two will try finishing you off, giving your team even more kills.

Late Game:
You'll be decently tanky, and deal lots of damage.

Later Game:
If the game lasts too long, the enemy team will probably start to hurt you more. Fortunately, the same goes for your team. So you need to be able to act quickly in teamfights and make sure that you do your part because the fight will get real ugly real fast. This is when the Defensive masteries really help, when every second of survivability matters.

As the tank, you will notice that enemies will try to ignore you and focus someone else. Therefore, you need to give the enemy team reason to try to kill you.

Be a **** to them in teamfights. Keep Bandage Tossing carries, button mashing 'E', turn on Despair. You'll do a lot of damage for a tank, so they will have to do something about you or they'll probably lose the teamfight. Not to mention Curse of the Sad Mummy which is a 2 second "insta-stun" on everyone within the radius.

Try to maintain jungle vision. Ward jungle entrances and Smite Wolves for extra vision.

Push lanes. Because you're a tank, even if 2 or 3 people are MIA and try to gank you, they can't. Of course, this doesn't mean that you shouldn't be careful at all. The key is to be both bold and wise.

Steal their jungle.

Take Blue Buff if it's for the best. If your Ally mid is Manaless, or they're just unbelievably far behind where Blue would just be taken away by the Enemy mid, then you should take it.

When pushing lanes, run ahead and take the minion aggro for reduced Tantrum cooldown.

Don't forget to ward, and ask your laning partner to do so as well. It could save your life!

Try to get minion agro when you're approaching an enemy, whether it's for a gank or a teamfight. Minion agro = lower Tantrum cooldown.

Use Curse of the Sad Mummy when your team is ready. This is how I do it: Bandage Toss in there, Tantrum and Despair, enemies attack you since you're right in their face, your allies start attacking enemy since you're being focused, now use Curse of the Sad Mummy. It won't always be like this, but a lot of the time, it will.

1. Button mash 'E' in team fights
2. ???
3. Profit

Don't forget to use your passive, Cursed Touch

Sorry these aren't all Amumu matches, but you get the idea.