This is by no means an be-all end-all list for this build. This is just what my items usually end up looking like when I play Amumu. First, I figure I will have to explain some of my choices.

Merc Treads - This is what I buy 90% percent of the time, only replaced by Ninja Tabi when there are mainly AD champs and little crowd control on the opposing team.

Rylai's - Health and AP is a huge plus right off the bat. Plus, combine the slowdown with a few well-placed bandage tosses and no one is escaping you, meaning more successful ganks.

Abyssal Sceptor - Lots of AP, lots of MR. Against a team with even 2 AP champs, this is a must for this build.

Shurelya's - I know its common to chunk gold items once you've stocked up, but having extra health, 15% cdr, and a speed boost for team escapes makes backdooring and deep pushes all the easier to get out alive from.

Rod of Ages - Why is it last? Because its my trade-away item. End game, all I look for is what will make things easiest for survival, so the four items that could occupy my sixth spot are Rod of Ages, Thornmail, Lich's Bane, and Banshee's Veil.

Rod of Ages comes out most of the time when all I'm looking for is extra health, mana and AP. To me, RoA stands on its own, even without getting the full buffed form.
Thornmail should be self-explanatory; it keeps the AD auto-attackers dead and me alive.
Lich's Bane comes in when either I'm having trouble chasing down people, even with rylai's help, or I feel only need some extra MR to go with my AP.
Banshee's Veil should also be self explanatory; when I'm not staying alive long enough to do any damage, I don't need to invest in extra AP. If it's bad enough, this item comes in to slot 4 or 5.

Pros:
- Higher damage output; to the point where I can usually 1v1 most champs. Even a little AP will spike your damage with every ability; its very deceptive when a tanky Amumu can do damage. The extra 1~2% max hp/sec on despair really makes a difference.
- Rylai's keeps even the faster champs from escaping without the use of flash/ghost/etc.

Cons:
- More squishy than a normal tank build. If one person dies in on my team in a group fight, its usually me if the rangers know to keep their distance.
- Low max mana. Must be conservative early game or recall quite often.

I tend to play with junglers a lot, so this build was made primarily to allow me to solo-lane effectively; even though the heart of it comes after I should already be done laning.

Basically just grab the abilities in the order shown and minion feed like you normally would. Be a little conservative with how much you use your abilities, because like mentioned above, you dont have a lot of mana to work with, and clarity, pots, and phil stone only go so far.

If you feel like you wont be getting many kills/assists, grab a second heart of gold so you can build up extra health and money.

Now once you buy that nifty Sunfire Cape, the fun is about to begin.

Once you get Rylai's and Abyssal Sceptor, congratulations, you are now officially a walking death aura. Don't get too crazy; you're not invincible like a normal tank. That aside, you should still be in the center of the fights as much as possible, cause otherwise, all that AP just goes to waste.

Just play Amumu like he's meant to be played and have fun racking up 7-15 kills and 15-25 assists per game!