Amumu, in my opinion, is one of the best champion to fill the jungling role despite being so easy to disrupt. Although he easily counter-jungled, he gets a massive trade off mid and late game. As Amumu, it is your job to soak damage, disrupt the enemy team, proc your debuff, and protect your carry. The last part isn't as intuitive but I'll get back to you on that later.

If you start with you have to start on Ancient Golem with a leash. Follow on to Wolves, Wraith, Double Golem, and Ancient Lizard. Now you should be level 4 and near full health to gank. Look for ganks top, mid, and bottom. You can always just attempt a gank you know you're going to fail to just burn their summoner's spell and apply pressure. If the jungle is respawning, get back to it ASAP because you want to get to lvl 6 because your gank potential is so much higher then.

If you start with you should do Wraith Camp first (smite the big 1), then wolves and then wait 5-10 seconds to do golem. It is best to just run back to mid last for a few seconds to avoid clairvoyance then get back on to ancient golem. At level 2 your should be able to solo Ancient Golem with relative ease. You want to hit the Ancient Golem when your smite timer is at around 12 seconds because thats how long its going to take you to kill it.

You can now track back to double golem, as you finish double golem, the wraith camp should spawn then so should the wolf pack. You would now go to base, pick up several more heath potions, defensive items, and go back to lizard and double golem. By now you are nearing 6. You should look for gank if you can, otherwise get to level 6.

In team fights, you want to position your self so that you are in front and your allies are in a cone behind you. Make sure your range is about 600-900 range behind you and your melee is almost side by side with you when you are roaming. Being close means that your allies can follow up immediately when you've caught someone. Having your allies cone our slightly prevents them all from being hit by aoe and cc's by flanks.

As Amumu, after the initial engage you want to tantrum and despair but delay your ultimate until you see the enemy starting using their burst on either you or an ally. You can either use your bandage to single out 1 person if they are all running, cc the person zerging your carry have you use your ultimate, or cc their carry.

A lot of people ask me what the best junglers are and I usually say Amumu or Nunu. When they ask whats the absolute best, my best answer would be, "it depends." It depends because Amumu or Nunu works better with different team comps. Champions that works especially well with Amumu are AOE heavy Caster carries.

Champions that can fully take advantage of & are:

Annie, Vlad, Brand, Corki (heavy magic component), Karthus, and Miss Fortune