So, you want to play Amumu. Or, you already play Amumu, but aren't sure what to do next. Support Bot isn't doing it for you, or your jungleing and ganks just aren't where you want them to be.

This guide is for you. If you have an anyway half decent team, and can land skillshots regularly, this build essentially guarantees end game victory. Amumu's natural tankiness combined with powerful CC and AP scaling will allow you to control the game with ganks, counterganks, initiation, and cleanup duty.

Amumu in the jungle is best off starting incredibly well armoured. It's the armour runeset that allows him to start with health regen instead of the standard jungler cloth and five pots build, and still come out on near full health for a gank. For a little more endgame survivability (Not that it's really required), you can use the percent health runes.

The magic pen marks are essential for this build though - They increase your damage output early in the jungle, in every gank you will ever make, and all the way through to end game. With Amumu's skills being mostly AOE, this small amount of pen quickly translates to a lot of damage to the opposing team.

The only change that I would suggest to the main rune build of Armor (apart from the %health) would be to use magic pen quints instead of armour quints. While this will slow down your early game slighty, it results in a much higher damage output over the course of the game.

There's no alternative here. You're amumu. You're built to be tanky and take damage. However, with this build you give just as good (If not better) than you get. It's for that reason you build the into the cooldown/ 10% magic pen tree.

If you were to adapt this build for a support/lane amumu, you could potentially change the final 9 into the support tree, just for the extra mana regen.

You see those items up there?

Get them.

Hold onto your GP5 items until you absolutely can't. They make up for those failed ganks when teammates aren't paying attention/plain mess up.

The main variation for the item build is the order in which you get things. DO NOT VARY BEFORE SUNFIRE. Except for maybe finishing boots.

For AP heavy opposing teams:
Get Abyssals before Rylais, and potentially get a Force of Nature as well in place of Rabaddons if you keep getting dropped.

For AD heavy opposing teams:
Just follow the build. If you must, you can grab thornmail after sunfire, again sacrificing Rabaddons.

For CC heavy opposing teams:
Who cares? You're amumu. If they drop everything on you, your team has free reign to destroy them.

For Extremely Long games:
Mobility is going to stop mattering: Everything is going to resolve into teamfights which you will either win or lose. So drop the boots, and pick up something else. If you're surviving mostly, grab a void staff for the extra pen and damage output. If you need a bit more survivability, grab a warmogs at the cost of your damage output. The last last item is purely situational, so use your better judgement

Follow that sequence. It presumes a standard jungle pattern (Blue->Wolves->Wraiths->Golems->Red->Wraiths->Gank)
the second wraiths lands you on level 4, and if you've done your gank well, you're going back to base and coming out with your Philly and boots.

Always max tantrum first- it's your most consistant damage source for both jungling and ganks. You max toss next for the CDR, and tears last because all it increases is %dmg, which is most effective end game.

If you must go for an early gank, bandage toss can be grabbed at level 3.

IMPORTANT:
Even if you're going to try and steal their blue, or invade, or whatever, don't get bandage toss first, get Tantrum. While the bandage toss stun is handy, you will fall behind your jungle pattern and in levels for the first couple of minutes.

Pro:
You're more or less unstoppable all game.
Early Tankiness carries through to late game very well
Damage output unexpectedly high to most opponents
Abyssal stops people raging about "Not being support tank"
When playing with subpar carries, your damage output more than makes up the difference (And if they're good, then you roll through things)
Teleport combined with increased damage output through mid game negates or turns around most ganks.

Cons:
Slightly less tanky than a normal amumu
Fewer support items as a trade off for increased damage & Utility
Early ganks depend upon your teams laner too much (true of any amumu)
Dependant on blue buff or lots of mana potions

This Build exists for those times you cannot truly trust your team, or you just want to force end game team fights in your favour. If the enemy team focuses you, GREAT! That's your job. If they don't, they die. Quite simple really.