Amumu seems incredibly strong late game, and I was just wondering exactly how people see him in the current meta. He's prone to invasion early, but can really carry mid/late game. How does he work out in the various elo ranges?

Ever since the skarner nerf, I think he's incredibly powerful at all elo's. As long as you don't get counterjungled early (which is hard because of good warding and such), you're pretty safe until 6. Not only do you carry super hard late game, you don't need that many items, which fits great into the meta. Also, good aoe clear, not THAT reliant on blue.

TLDR; Amu is good. Play him.

If his team gets ahead, he'll generally ensure they stay ahead. If his team's behind, he can turn the situation around in a team fight, but that's not guaranteed. He's generally quite good with a competent team, so at high ELO, he can be downright scary, but at low ELO, teams that don't make adequate use of his enormous team fight presence can lose when they try to play assassin style. Also, he tends to be a little lacking in the 1v1 department late game so your game play as a very fed Amumu would be different from your game play as say, a very fed Jax.

Tldr: I've seen people catch up thanks to Amumu. I've seen people maintain their lead thanks to Amumu. I rarely see teams that initially win but then lose with Amumu unless it's one of those 50+ minute games where ADCs on either team take people out in 2 seconds. He's very strong with a competent team, especially an AOE team

I use to main him in season 2, before he became popular and got nerfed and even then, I played him often. I'd drop about 10 pink wards a game and still have a huge impact in team fights.

He's banned always for a reason.
The dude is AWESOME
and he's FUN to play.

Tip- if they invade your blue, run quickly to theirs with your laner. Dont try to fight it. If that isn't an option just clear what you can and go top for a gank early. That early xp/gold will make up for it.
Amumu lvl 3 ganks are OP. you can do it lvl 2 as well if you run exhaust or ignite instead of flash, but you really want the flash ult option.
One time I was feeling a little bit trolly So I walked up to enemy turret at minute 1 and spammed my taunt and laugh. I then let them chase me for a while and they got FB (our mid also got a kill, irrelevant).
My team raged at me until 5 minutes later I had 3 successful ganks on top and mid. I ended that game 13/1/18.
Game ended and the enemy team was swearing in all chat about smurf accounts...

I took it as a compliment and a tribute to the fact that I'm a halfway competant player.

He's one of my favorite champions to play. People simply cannot leave him unchecked in a match or it just makes it easier for him to help his team get ahead.

He has one of the highest win ratios in the game atm, and for good reason.