Alright, this is the build I have been using for Amumu for some time now with great success. Overall, this build will provide your team with: Extra time for growth, due to the early Jungling; A Tank that takes damage from all five enemies for at least six seconds on retreat; A Nuke that can initiate any team fight; And a Pusher that can slice through minions like it's his job.

As of now, I go flat Magic Pen for early game jungle and late game damage, flat Health to maintain survivability in jungling, Ability power/level for that late game nuke, and two flat Magic Pen for increased damage output and one flat Health

Offense: This is a no-brainer. You are going to jungle, so buff smite.

Defense: Basically, you are working towards survivability both early and late game. The four points in the Scars makes the jungling a snap and to get there, you get the wonderful bonuses of taking less damage (Harden skin, Defensive mastery), regaining health faster, and increased armor and magic resist. I HIGHLY recommend not going for the increased dodge chance masteries here, taking hits with Amumu helps more than it hurts with his Tantrum ability.

Utility: Speed. Being able to get through the jungle quickly while maintaining a level equivalent to a dual laner. Therefore, hit up the health and mana regen, recently, I have been play testing with teleport in place of clarity therefore one point there, if you dont use it, dont waste the point. Then rush the four points in exp boost. Past that, the two points in the neutral buff extension are non-negotiable, but the single point in greed is pretty flexable. Greed is great, but with the item build, you shouldn't have much trouble getting gold.

Cloth Armor and 5 Health Potions? Jungler.

After first lap through the jungle, start the philosopher's stone with the Regrowth Pendant over the Faerie Charm because you should still have the blue buff. Grab two or three more Health Potions. If you are not running clarity, you may want a Mana Potion, if you are, dont waste the money.

Finish your second lap and grab the philosopher's stone and at least the Boots, hopefully the full Boots of Swiftness.

At this point, the Heart of Gold. Then you should be ganking or hittin up the dragon.

The Catalyst the protector is extremely helpful once you get into ganking. You should be popping out to grab someone then clearing out the minions, so your levels should be moving pretty quick, this essentially allows you to recover from a gank without having to base often.

Banshee's Veil. Great item, very crucial against a team with alot of debuffs, the added health is always a plus. Situationally, this can be switched for other things, such as a Quicksilver Sash to deal with Warwick or Malzahar. Another good option is grabbing a Rod of Ages if you find your team needs more damage output [this option allows you to keep the benefit of the catalyst]

Sunfire Aegis: stereotypical tank item, with Amumu's Despair he is packing two sources of passive damage over time...ridiculous

Rylai's Crystal Scepter: Your tears now slow the opponents, congrats. Also, the extra Hp never hurts

Shurelya's Battlesong: Cooldown reduction and health bonus are enough reason to grab this item, the active on it however means that your enemies wont be running away from you or your team any time soon.

Last but not least: Randuin's Omen. There will always be times where you will have to run, this makes it simple. The scepter is already keeping anyone too close slowed, but now you have a 20% passive chance of slowing your enemy...when they hit you...and when you pop its active, you shouldn't have much trouble out running most any champs.

----Other Items----

Quicksilver Sash. This item gets you out of suppress and other hard CC. Other than that, the MR it gives is insignificant compared to the passive of Banshee's Veil. Get this if you are up against a good Warwick or Malzahar.

Rod of Ages. If picked up early enough, scales Amumu into a hefty nuker. Swap in for Banshee's Veil if you team has enough tank.

Rabadon's Deathcap. Swap this in for Sunfire Aegis to make Amumu's burst crazy, but only in the case of a low AD opposing team.

This item build is much harder to use, but results in a tank who can effectively leave targets open for a carry's assault. I came up with this idea right after the first change in Randuin's Omen and Heart of Gold.

Start with Meki Pendant and two Health Potions. Take blue buff and wolves and return to base. Buy three more Health Potions and either teleport to mid tower or walk to wraiths.

After the first lap through the jungle, pick up Regrowth Pendant and head back to wolves, wraiths and golems. That should get you enough gold to pick up Philosopher's Stone.

Boots of Swiftness. This item on Amumu allows his Bandage Toss to be effective, and increases his overall ability to initiate.

Banshee's Veil. Tank item, same reasons as above. This build caters well to Rod of Ages as well.

Sunfire Aegis. Armor, doubles your AoE DoT.

Rylai's Crystal Scepter. Amumu crys tears of ice and his opponents are frozen with pity. Extra HP and AP never hurt.

----Other Items----

Quicksilver Sash. This item gets you out of suppress and other hard CC. Other than that, the MR it gives is insignificant compared to the passive of Banshee's Veil. Get this if you are up against a good Warwick or Malzahar.

Rod of Ages. If picked up early enough, scales Amumu into a hefty nuker. Swap in for Banshee's Veil if you team has enough tank.

Rabadon's Deathcap. Swap this in for Sunfire Aegis to make Amumu's burst crazy, but only in the case of a low AD opposing team.

Despair: always lvl 1, it's how you jungle.

From there, I like to max out Tantrum as soon as possible in order to speed up the jungling while being able to pack a punch in a gank.

Lvl 4 Bandage Toss is crucial, at this point, unless your team is destroying the other, someone will prolly need a gank. you will be either on lvl 5 or 6, allowing you to stun, essentially winning the gank.

Amumu is a nuke, up his ulti every chance you get.

After 6, prioritize in this manner: Curse of the Sad Mummy, Tantrum, Despair, Bandage Toss. Once you have a point in the toss, the added damage and faster cooldown is trivial, you get more benefit from the others.

Best Choices:

Smite: you are a jungler.

Teleport: Your team is in trouble up top and you just killed the golem duo, this is just the thing.

Clarity: A good choice, but there may be others, this is typically where I go just because it makes running through the jungle pretty easy, even if you happen to lose blue buff too soon. This spell becomes a lot less useful late game, esp. if you are maintaining your blue buff. I recommend getting the hang of the jungle procedure, then weaning off this spell.

Flash: Great for survivability, a nice initiating tool, flash in - Curse of the Sad Mummy - team fight. Takes some getting used to and i wouldn't bother upgrading.

Fortify: Not all that helpful by itself, but never an entirely wrong choice for a tank.

Other Options:

Ignite is great for ganking, but you shouldn't be letting them get away.

Ghost is nice when trying to return to battle, or get out if and when the other team doesn't focus you and your team is dead.

Fortify Useful, only use if you have good map awareness.

Bad Choices:

Revive. Honestly, kind of a waste of the spell slot.

Exhaust. Amumu has two hard CC abilities and gets a soft CC late game.

Overall, you will end up with a great champion who can survive, and can block a whole team when necessary. The biggest part to learn is how to get through the jungle without being a detriment to your team, after that, learning his abilities' idiosyncrasies and how and when to set up a successful gank or team fight are the only things that stand in your way of wielding this great champion.

For those of you who are not familiar with jungling on Summoner's Rift, I will take you through, step-by-step.

1. Buy your starting items and walk to the Blue buff Golem spawn. (The spot with the big golem and two small creeps)

2. At 1:55, the golem will spawn, if you are afraid of being ganked early, have a team mate stand in the river to guard. Attack the golem and toggle Despair to ON. Immediately, use your first Health Potion, this should last through the fight, but be ready to use a second. Make sure to use Smite on the Golem, if you don't you will almost assuredly die.

3. After the golem is dead, kill both of the creeps and you should be on lvl 2. Remember to toggle Despair to OFF

4. Use your second Health Potion if you haven't already and walk to the wolves. Attack the big wolf and toggle Despair back on. Tantrum should be used twice in this fight, more will cause you to be mana hungry through the rest of the jungle. The two uses should kill the big wolf. (continue this fighting style while jungling, esp. when you have the blue buff) When the wolves are dead, toggle [[Despair] to OFF.

5. Walk over to the wraiths (across the mid lane), toggle Despair, and attack the large purple one, all four should die approx. the same time.

6. If you have all of your runes and masteries, go for the lizard (red buff). If you are still working on this build, you may want to skip to the small golems for the first lap. (Dying is pretty much the worst thing you can do)

7. By this time, your Smite should be off cooldown so use it on the big Lizard, and attack him, your Tantrum and Despair should kill the others.

8. Lastly, go to the small golems near the lane, they hit hard early, but have small amounts of health.

That is one successful lap of jungling. You should be around lvl 5 by now, so after basing to buy the regen pendant and more hp pots, ask if anyone needs a gank. If not, head to the wolves and restart (Blue golem if it has respawned already). After both laps, you should be between lvl 7 and 9, if you feel confident, go for the dragon with at least two potions and smite, if not, gank.

[NOTE: Taking the dragon at lvl 4 is entirely possible, get a feel for the fight and make sure you have both buffs...it is risky if their team has a lot of map awareness.]

From here on out, grab the buffs when you aren't busy and press in the lanes helping out where your team needs you.