Welcome to my jungle amumu guide! To start with , lets go with jungle routes.
First you want to have a pretty strong leash at blue (Do not start wolves , you'll end up taking to much damage before you get tantrum) So blue --> Wolves --> Wraith --> Golems --> Wolves --> wraith --> Red --> golem --> By this time you should be level 5 , Gank lanes if you see good opportunities , when you have enough gold go back to base , get a philo , Heart of gold and a pair of boots (If you can afford it!) and go back , get your blue and gank again and again :) You are tanky , you do tons of damage , you have tons of CC and you can roam like the mummy you are!

You start with a regrowth pendant and a pot , with the tanky masteries and the armor runes this is enough for you to clear the jungle and have enough HPs to gank. When you have enough gold (By the time you hit level 5 or 6 , depends on how many ganks you did) go back and get a philosopher stone , heart of gold and boots. Gank and farm , Now go get mercuries thread's and a sunfire cape (Sunfire and your W / E combo will eat their hps , that mixed with your Q / R will kill anything that tries to run away) now that you have alot of golds , go back to base and start to build your frozen heart starting with a glacial shroud (mana + defence is always nice!) and if you can grab a negatron cloak.You are now one tanky mummy.Now go back to do your thing, jungle,dragon,teamfights and ganks. You now have golds! (The gold items and the gold quints make you farm sooo much) go back and finish that frozen heart and a FON (banshees is good but the force of nature is just better IMO. If you want to , you can always get a banshees veil later if they have some heavy magic damage (Remplace the radiums OR shurelyas by banshees) , if they have more auto attackers grab a thornmail:) Go farm some more and build that heart of gold into a radiums omen and that philo stone in a shurelyas. That's about it for the item build! You now have mercuries thread , Sunfire cape , Frozen heart, Force of nature , Radiums omen and Shurelyas/Banshees/Thornmail!

Well this chapter's gonna be short. Flash/Smite. Nothing else is really needed.
Flash - get those nasty flash/Q/R ganks , Smite well you're jungling.
You could also go Ghost/Smite but flash is just the best in my book.

Well , Runes..
Mpen reds , I find them to be the best for amumu. Some extra damage from all of your spells. ^^
Armor yellows , gives you the early game armor you need to jungle without having to heal every time you clear. I wouldn't change them for anything
Magic resist/level Blues , My build early game involves very little magic resist (only mercuries thread before I get a FON.) so magic resist runes will help you alot to survive while ganking.
Quints.. gold per 10 is what I use , the items you need to be affective are pricy , gp/10 are really good on amumu.
You can also go for mspeed/quints , losing a kill because that one guy with 5 more movement speed is frustrating , the 4,5% mspeed bonus mixed with FON will help you chase better. :) I preffer gp/10 over them anyway.

There is not much to say , you will not be able to gank before you hit level 5 so you need to tell your lanes to be careful. Once you hit 5 or 6 your ganks are really scary , maxing E first really help you, then W , R and Q.

0/21/9. Well you are one tanky mummy, so going 21 in defence is what you need
That's pretty self explainatory I guess. Smite gold bonus , armor and mres , Reduced damage from minions for the early game , health , health regen and CDR . Amumu needs all of it.
And 9 in utulity.. He is SOOO mana dependant all game long so the 20% extra duration on the blue buff is a must , movement speed is very nice too , extra mana/level helps and the improved flash can save life , or help you grab a kill!

Pro's
-Clear jungle really fast
-High ammount of CC
-Tanky with a good ammount of damage
-Almost a guaranteed kill once he hit level 6 and gank
-Look sexy and has a sexy voice.
-Can control a team fight with his ult.

Con's
-Heavily team dependant
-Can't gank pre-5
-Always sad
-Hard to master , His Q animation is slower than most skillshot so it takes practice and you need to time your ULT perfectly to make the best use of it.

amumu need to have a team to win , obviously.
Early game ; You will need to gank alot once you hit level 5 , help your carry to get fed by ganking like a boss. this amumu build makes him a great tank , once you get your Sunfire cape don't be afraid to tower dive to get some kills and feed your carry.

Mid game - Now that most of the first turrets are down you will need to roam , ward and gank the over extended enemies , don't be afraid to engage ,you have alot of armor (215+) and about 150 magic resist , and a tons of HP , Steal the enemy jungle , ward and catch your enemy in bad positions and make good use of that oracle .

Late game - You now have 300 to 400 armor , 200 to 250 magic resist and 3200 hps , you are the tankiest champion in the game and you do tons of damage. Engage in tower if you need it , win team fights by CC'ing their whole team and win games.

This end the guide.. Amumu is a really fun jungler to play, really strong from early to late game, His ultimate is amazing and he can really help . He is not the highest damage jungler but once you know how to play him well and how to get those Q's on he become one of the best ganker in the game.Feel free to leave comments and/or to rate it. If you want to play add me in game , Jenifael. I am always down to play a game and to prove you that this build works like a charm. :) I know i have made many grammar mistakes , please grammar nazi forgive me, I am french (inb4 whyGuideNotIncludeSurrender!) so english is not my main language :)