Amumu is a great jungler and one of the strongest tanks out there, ive therefore made this build to help people getting to know this bipolar-emo-pharao a little better.
ps: this is my first guide so any good ideas and constructive critsism is welcome:)

Pros:
Very strong tank if played right.
Extremely good ultimate.
Strong initiator.
Good jungler.
Good AoE damage(especially vs high-hp champs)
Farms minions like a madman:)

Cons:
Needs blue buff at jungle start
Requires a good team behind him.
Some learning curve at his Skill-Shot/Ultimate usage
Not as strong a ganker as nocturne/warwick/shaco

Ill skip the runes and masteries section since this is self explanatory.

Start out with a Cloth Armor and 5x health potions, go straight to the blue buff and make sure to have people scouting for you because its vital to amumu's jungle route.
when you have jungeled up to 600 gold you blue pill, sell your armor and buy Philosopher's Stone.
This item is one of the most underestimated items in the game, for relativly little gold it gives you insane hp/5 and mp/5.
If you are however not jungling skip the armor and potions and go straight for this item since it lets you lane nearly indefinetly without having to go back to regain health and mana.

After this you want the Mercury Threads for the + MR and cc reduction, since we'll have added movement speed from both masteries and force of nature the Boots of Swiftness is just not viable as an item choice.

Your next item should now be a Giants Belt for added HP, you might ask why i pick this before i make Force of Nature and the reason is that it helps you utilize FoN's passive effect and gives you the health boost needed for ealy-midgame tanking.

After FoN complete your Sunfire Cape, even if the enemy is all AP this item gives you a good health boost and added AoE damage wich works well with your other abilities.

After you complete Sunfire Cape things become alot more flexible and here is why:

- are the enemy team mostly beefy AD (Sion, Jarvan, etc.) then build a Randuin's Omen
- are the enemy mostly squishy AD (Tryndamere, Master Yi, Miss Fortune etc.) get a Thornmail
- if the enemy is heavy AP build a Banshee's Veil or Abyssal Scepter if your team is doing well, since this item lowers your enemys MR it gives you better damage.
- if your own team is composed of alot of squishie high dps champs id recomend to buy a Aegis of the Legion since its aura effect increases your teammates MR and armor, it might not seem much but it matters

Note:
You should also remember that alot of champions have mixed damage(mundos cleaver, poppys charge, ezrael, jax and twisted fate.) and you might have to build both MR and armor to counter them.

Beeing able to adapt to a game as a tank is vital, even if the enemy is composed of mostly AD chars but their Lux is doing well and getting fed then its worth to invest some extra gold in some added MR, and same the other way.

This starting build have mostly MR items in it mainly because of Force of Nature, it is however a very good item to have vs all types of damage mainly because you now have massive HP regen, this lets you go in first and tank run or flash out at low HP then wait a few seconds and jump back in. it also keeps you from going back too heal most of the time, meaning you can fight or farm nearly the whole game with only going back for items and pots.

Amumu is a bit hard too learn at the start, when to insitiate a teamfight and when to not, its vital to time your attacks against the other team and try to catch as many people possible with your ultimate.

The ultimate has an understandably long cooldown and you should always have the blue buff to reduce this as much as possible, if someone else is taking it build Shurelia's Reverie from Philosopher's Stone when you have enough money since it gives 15% cd reduction or vs AD team rush a Randuins Omen.
Frozen Heart is also a good item for CD/armor but i personally dont fancy it over RO.

Jungling:
Your jungle route should be Blue buff -> Wolves -> Wraiths -> Small Golems and back to buy items, after this kill whatever you want and be sure to kill the Dragon around lvl 7-8.
You should also try to gank around lvl 4-5 and over to help out the people on the lanes, always say or ping the target you want to gank so that your teammates know you are comming and can prepare.

Always try to signalise your intentions when initiating teamfights and ganking since it helps your team alot, ive seen many who just rush in and are left stranded and dead because their team wasnt redy.

One thing to remember is your passive, each time you hit someone you reduce their magic resist for a set amount, so always auto attack the target your team is focusing.

Another tip if you're struggling with his Bandage Toss is to go into the Menu -> More Options and turn of "Show line missile display"

Well thats it for this time, i dont know yet how to incorporate pictures and links yet so if any1 know plz comment and tell me how:P

I've also tried this build on Galio with good results but without the jungle part naturally.

hope this build helps you out and plz comment, critisze and share your ideas.

-Lazsaron