Very little has gone wrong with this car so far.
The day I got the car (it had 47095 miles on it at the time) the turn signal relay went out, but the dealer fixed it and it hasn't been a problem since.
At 56000 I had a minor wreck and hit the front bumper. 2 weeks later the power steering pump went out. I'm not sure if the power steering failure had anything to do with the wreck and fixing it cost about $200.
The front driver's side seat had a 4 inch rip in it that I sewed up myself. It looks like brand new.
I did an oil change at 54000 miles and recently, about 3000 miles after the oil change, started to hear the engine clicking occasionally. This could be caused by cold weather, or from the weight of the oil I used (5W-30). I'm also told the wreck could have knocked a bolt lose in the motor and that maybe responsible for the clicking. I'm going to keep an eye on it and see if it gets worse.
Overall, this has been a great car. It's got very good safety features, and handles great. It's a lot of fun to drive! And for just being a 4-cylinder, I think it has a lot of power, especially when first taking off.
Also, a lot of people have complained about brake problems with these cars. I've only put about 10000 miles on this car since I've had it, but so far the brakes are still in perfect shape. The previous owner of my vehicle also told me that she never had any brake problems (or other mechanical problems) with this car.
There have been minor things that have gone wrong with it, but nothing major (not yet anyway). I have a high opinion of GM cars and have been pretty impressed with my Alero. I recommend them to people who are looking at buying a new car.