Amumu has quickly become my favourite tank, I have tried out many varying builds and have settled on this one as a default build.
Now as with any tank you should take the build as a guide and build what is needed to counter the opposite team, either by prioritising certain items or by switching them out completely.

This build focuses on mass crowd control with enough damage so you cannot be ignored whilst maintaining survivability.

Tanking is all about reducing damage given to your fellow team mates.

It is NOT about having so much health/armour/resist that you cannot be killed but can be completely ignored by the other team, its about forcing the other team to focus on you so that your team can unleash their full potential on them. So you need to make your self an inviting target but be strong enough to survive a while, and this is how i tank.

Pros
Very large amount of Crowd control
Surprisingly good sustained damage output
Team focused tank
You will pick up many assists and a few kills by accident
Feared by many
A good initiate will secure a win in most team fights
Good income from gold items and mid game farming
Global mobility with teleport
Good sustainability due to regen items
Good warding character (I buy 15-30 wards most battles)
You do not need to be fed to be effective but a fed Amumu is damn scary
Despair tears through high health enemies

Cons
If your team falls FAR behind you will struggle to tank effectively
You need to be careful at start
Often banned in ranked

Tip
Flash can enable you to reset a bad initiate

so what does Amumu bring to the table to deserve the title of King of CC

1) His ult gets him the title of team destroyer and is why Amumu will often get banned in ranked as causing the opposite team to stand still for 2 seconds can completely destroy them.

2) Bandage toss is a good initiator and also apply's a short stun which when well placed can come in very handy in chase/flee situations.

3) Randuin's Omen brings in a area 35% slow for not only movement but also attack speed and also the same slow for any one who attacks you.

4) Rylai's Crystal Scepter is the weakest of your CC only really bringing in a 15% slow for anyone in your area of Despair and a very mild extension to you bandage toss stun.

The 4 forms of CC will cause havoc in any team fight but perfecting when to use what CC can take a little while to master.

Its best to be paired with a ranged character as if your facing ranged opponents it can be difficult to farm effectively without taking large amounts of damage in the process. This often leads you into a roll of harassment and ensuring your ranged partner gets as many last hits as possible during the lane phase. Whilst this will boost your ranged partner in terms of gold you will often suffer, so the start of my build focuses on keeping us from falling to far behind.

To start with Rejuvenation Bead, Faerie Charm and a Health Potion should keep you in lane for a good while usually long enough to get Philosopher's Stone which will boost your regen ability and hopefully keep you up on gold.


Boots are a must next as you need to be mobile to be effective, and don't forget to buy wards each time you return to base 75 gold is well worth the gold to ensure an attack doesn't turn into a gank.

Next we need to toughen up and keep going with the gold so HoG makes total sense and build into a very good item later on.


I like to get Mercury Treads next as the resist and mobility as well as the CC reduction are essential to your build but for low CC but high damage teams it could be viable to go for Giants Belt first, either way you really need both before team fights really start cracking off or you wont last long in the thick of the fight.

To survive early game you will need to use Flash wisely and initiate with care, its much better to wait a few secs to get a good initiate on the main damage dealer than to run in head first and get beaten down.

The main items are

Randuin's Omen
Abyssal Scepter
Rylai's Crystal Scepter

and each game will call for them to be build in a different order

very generally speaking I go for Wardens Mail and then Abyssal Scepter before finishing Randuin's Omen and Rylai's Crystal Scepter. These 3 items however are interchangeable and should be build to best counter the opposing team.

Item Highlights
Randuin's Omen
2 forms of CC, loads of armour and some cooldown this basically is a AD carrys worst nightmare and the Active I class as a mini ult on a short cooldown.

Abyssal Scepter
A good amount of Resist combined with AP and a aura to reduce resist. basiaclly a very good item for your build and a very good item to assist the AP chars on your team.

Rylai's Crystal Scepter
A small health boost and a large AP boost as well as a Passive to turn your Despair ability into a nasty little slow.


At this point you should have 150 AP which will make your ability's hurt that much more, and unless there AD carry has brought defensive items you should be doing true damage.
Bandage 320 + 150
Despair 24 + 2.7% + 1.5%
Tantrum 175 + 75
Curse 350 + 150
= 1220 with 24 + 4.2% each second when there near you which is rather devastating to any squishy character especially when considering your stuns and slows.

So at this point I would like to point out the importance of a good initiate as you can destroy a squishy damage dealer removing them from then fight via death or forcing then to flash away etc or tickle a tank and leave a carry to tear into some one.

Late game items really do vary wildly and should be topping up in a area that's needed i put the standard as Aegis of the Legion as this will bulk up your tank roll and Shurelya's Reverie as even the king of CC needs the odd speed boost.

Really though keep the Philosopher's Stone as long as you can and here is a quick run down of my usual items

Aegis of the Legion
General tank item useful for teams that get stuck in the fight with you

Banshee's Veil
If counter CC and AP is getting to much for you

Force of Nature
For high AP teams

Frozen Heart
For hard AD teams

Guardian Angel
if your getting focused down, but be wary this item has a do not focus me effect and you are the tank you want to attract attention

Rabadon's Deathcap
If you can get away with being an offtank this item will add 487.5 damage to your 3 spell combo making you very deadly to any carry.

Shurelya's Reverie
small health boost but mainly used for a team speed boost and cooldowns

Thornmail
Not really a fav as Randuin's Omen should cover you on the anti AD carry front put a possibility in some situations.

Warmog's Armor
If you need a large health boost and the game looks to be lasting a while longer this can be useful

Zhonya's Hourglass
An excellent item to counter AD whilst giving you a large amount of AP and a stop damage button (very useful to counter Karthus)

As the tank its your job to get a good initiate or to at least screw up there initiate and Amumu is set up well for either situation.

Ideally you will bandage toss onto the AD carry or another weak but heavy damage dealer, active Despair on the way in and use tantrum to hit multiple targets and then ult the entire team, and once this wears off use Randuin's Omen to CC them even more.
Now timing is everything on this not only do you need a good initiate you need to be in a position where your team can rapidly follow you in, if you use your ult and no one is benefiting from the opposing team standing still then it was a pointless ult.

Insight marks and quints to make the most of your damage and with 34 magic pen you should be doing true damage to any AD carry that hasn't sacrificed a damage item for defence.

Clarity seals should keep you out of most mana issues although if the blue buff is going free it means you can leave Despair on.

Celerity glyphs to help out on cooldowns

whilst I would keep the Insight runes the rest could be changed depending on play style possibly to boost armour and resist.

Offence tree is used to maximise damage via 15% magic pen and a little extra AP and some help for cooldowns, you don't really benefit from anything else on this tree whilst i like to have Archaic Knowledge you could trade this tree for Defence if that's more your style

Utility tree assist with both of your summoner spells as well as mana regen cooldowns speed and XP, the main questionable ones are greed which doesn't really add to much over a game but I see it as a free ward every 12.5 mins so not really a game changer but nothing else really stands out more especially as I don't tend to use the buffs unless no one else needs them. and finally you could switch out Good Hands for Perseverance if you prefer that one.

After starting with Bandage Toss i go with the following order of preference

Ult
Tantrum
Despair
Bandage Toss

I go with this to maximise my area damage rather than the single target damage of Bandage Toss

I love Flash and Teleport on Amumu as they just suit him so well and i find it hard to trade either one of them out

The only one that could benefit is smite but only if you wish to jungle on him, Amumu is considered a Tier 2 jungler so he is more than capable in the jungle but i prefer him in lane. if you wish to jungle on him it may be worth looking at a different build as this one is in no way setup for it.

Save flash to reset a bad initiate.

Do not rush you initiate waiting for a good time is defiantly worth it, but don't be a chicken waiting for the perfect moment pick your high priority targets before the fight and hit who ever gets in the way.

Get as many people in your ult as possible.

Use the active on Randuins omen to extend CC effects after your Ult has warn off.

Ward a lot you have the gold to do so and the survivability.

Do not chase kills that is not your role, Stick in the team fight.

Farm when you can mid game, you have teleport to get you quickly back into the fight.

Alter the build with the late game items as your tanking needs specify.

When getting tower dived back off and bandage toss to stun them before walking to other side of tower and using your ult this should maximise turret damage due to stunning them for for 3 seconds and crossing tower twice, Randuins omen and flash can also prolong this.

Your ult will not stop a channelling ult but your bandage toss will.

In lane harass with Bandage toss Tantrum then go hide in the brush, once you get Philosophers stone you should be able to quickly regen any damage taken.

Early game is tough so take it easy and don't go face checking bushes.

Feed your AD carry early game (give them the minion kills)

Keep flash ready for your first proper team fight you will probably need it to survive.

Don't focus high health enemies but try keep them in your Despair area