Hi, I am a player from the SEA sever, playing mostly as a jungler at the start of season 4.
Amumu is a fun jungler to play with great jungle clearing speed. This champion can devastate teams with his AOE ult. It is one of my main champions and I feel that it shines strongly from mid-late game. As this is my frist time making a guide for one of my favourite champions-Amumu, I would appreciate it if you leave a comment or constructive criticism and feedback.

Pros:
-Fun to play
-His tears hurts
-Can probably 1v1 almost anyone at midgame(take note of mana).
-Very fast jungle clearing speed
-Has a stun, and an AOE ult which wrapes enemies
-Tanky, while dealing a decent amount of damage
Cons:
-Cries a lot
-Gets bullied early
-Item dependant
-Mana hungry(Don't give your mid laner your blue, you need it more than him early on)
-If you can't land your Q, you are going to be a useless jungler

I usually go with these hybrid masteries as I feel they are good for Amumu. All of Amumu's spells are based of AP, so you are going to need some ability power. He is also a tank, so he needs some defensive stats, which may not be too important if you want to build him to late game where he buys tanky items, but still recommended(Helps you survive the occasional counter jungling or tower diving). The reason why I put points into the utility tree is because of 'Runic Affinity'. Amumu really needs the blue buff or he would not be able to clear jungle camps. He also needs mana regen because he's one of the most mana hungry junglers, but extremely deadly when played right.

Smite

: Self explanatory, you're a jungler. I always take smite although it loses its usefullness late game as it could be used to steal the dragon/baron/buffs.

Flash

: Good for flashing into a team and ulting, escaping, flashing to get in range for your stun ect.

Teleport

: Not a bad spell, though I would prefer flash. It could set plays if your laners ward well or going for a backdoor, or teleporting to help your team during teamfights.

Exhaust

: A reasonable choice in place of flash if their team comp comprises of nukes, and it's also good for ensuring enemies do not get away.(You already have a stun and a root, now you got a slow!)

I usually max my E first to clear waves faster and get to 6 so I can gank with my ult.
I max Q next as it decreases the cooldown=more stuns and damage. However, be sure to aim it carefully.
I max W last as late game is when the skill becomes really useful. During late game, your W would be doing tons of damage to their tank and everyone in general. It also synergises well with Rylais.

Be sure to group and take the dragon(try to delay it early on, Amumu is quite weak without items), especially during mid-game, when you get a massive power spike with just 2-3 items.

Start from either blue golems, or the frog. Clear wraiths or wolves next(choose one), then go to golems with smite up for the stun buff. Take red, and return to base to buy your 2nd tier jungle item, or proceed to wolves/wraiths and then go back to base(If you have enough health).

I would usually go for the above mentioned, starting with Ranger's Trailblazer(Runeglaive), or stalker's blade if I'm ganking a lot. It is a really strong item, providing damage and cooldown reduction. One important aspect of it is the % returned as mana and health, which helps with your mana problems.
I would usually grab a giant's belt after my completed Runeglaive enhancement to build either into a rylais or sunfire cape. Sunfire cape, together with your W makes it painful for melee attackers.However, it can be substituted by frozen heart. Rylais has a slow which makes it a good choice, providing hp and ap as well.
After Your 3 main items: Runeglaive,Rylais,Sunfire/frozen heart, you can choose to get a Liandry's torment or banshees.
2 main ways to go:
If fed: Go the AP route with liandry as your 3rd item, and 1 defensive item(rylais does not count)
If evenly matched/losing: Get the tankier items first, with 1 damage item for clear speed.

Good players will build accordingly to the opponent's team composition and how the game is going. Having a guide just gives you a crash course about that champion, get the hang of it and you won't need guides.

Synergises well with:
-Katarina(Your stun+ult combo renders the opponent unable to do anything while kata ults and gets fed and snowballs hard. Game won. Kata is one of the best champions to have in Amumu's team as she does not require any mana, so you wont need to buy any mana pots)
-Riven(Manaless, can go mid, together you have 4 CC, can snowball)
-Tryndamere(Just stand there and watch while he hits your opponents once they are safetly wrapped up. His slow is also one of the most annoying spells in the game and should be nerfed. Together, its almost impossible for opponent to get away)

Gets raped by:
-Lee Sin
.Counter jungles you so hard you cant get to mid/late game.
.Only his eyes are wrapped up, while your whole body is.
.He can hear your cries
-Anyone with a spell shield
.Just when you land that perfect Q, boomz, negated.
-Olaf
.HIS ULT DOES WHAT?
-Yasuo
.Destroys your armor with his ult, crits every hit, has a shield that can soak up your Q damage, wind wall blocks your Q, can't gank that guy. RITO PLEASE!

Tips on how to handle those match-ups:
Ask for a hard leash.
Go to their red if they are counter jungling, that way, you don't lose out on buffs.
Ask your team to help ward red, in case of buff stealing, and ward red again when you are doing it. Feel free to call for back-up, that should land your team a first blood.
Blue buff is essential, DO NOT lose it or you will fall behind. Mana pots help, but they are not really cost effective.
When I am without blue, I usually do not toggle W on, and just use my E to clear. Get juggernaut if you are behind, and magus if you're ahead.

Try to predict where your opponent is going before throwing your Q, also, try to gank from blind spots.
If there is a counter gank, ask for your teammates to help, DO NOT risk dying even if you will lose your red buff.
Ward the opponent's jungle, always. I can't place enough emphasis on the importance of wards, if your team wards, the game is already half-won. With season 5 jungle, I will try to clear the river if I'm low on health, which also provides vision of that area.
Try to ult in the middle of a team, if your Q lands on the carry, just go all in and that carry will be dead before he can even flash(Never seen a carry taking cleanse while playing Amumu. For Nasus, but not Amumu? Wow, just wow).
If you run out of mana, go back to base, you can't clear your jungle camps without mana.
It does not matter who gets the kill, there will still be assist gold.
NEVER land an accidental Q into a prepared team when your team is not ready, prepare to ult and flash out.
Give the blue buff to the mid laner if you dont need it.
Don't be afraid to go all in after you have your core items, you do tons of damage mid game.