In my 55 years I had never hand an attempt made to pickpocket me until I visited Rome for three days. There were two attempts made at picking my pockets in those three days.
The first one was on a subway by two men. One of the men came up behind me and the other attempted to work my camera out of my pocket. My daughter who had been living in Europe informed me that the thief was probably attempting to get a hold of the camera strap and pull it from my pocket. I tried to move away but there wasn't room. So I turned, confronted them and suggested that I would appreciate it if they didn't steal my stuff ( I also made it clear that I intended them great harm if the planned to attack me).
The second attempt was on a subway also. A young girl that looked like she might have been about 17 put her hand in my front pocket and attempted to remove my wallet. Fortunately the wallet was solidly jammed in my front pocket and she couldn't get it out. She timed the attempt so that she and her partner were able to get off the subway just as the doors were closing. A local gave me a knowing look and suggested that this was a pretty common occurrence.
In addition to the two attempted thefts, our stop at the Trevi fountain was detracted from by a constant series of street vendors who seemed to be illegally harassing tourists. This was confirmed when a cop came by and chased them off.
Overall, I enjoyed Rome and would go back. My wife was sufficiently scared and annoyed that she says she'll never go back.
FWIW, I've visited major cities all around the world and it looked to me that Rome probably had the highest petty crime rate of any of them.
As for the PC comparisons of Rome with bad areas in the US in the above comments: I have spent a limited amount of time in bad areas in Los Angeles, Philadelphia and Washington DC. I never had a concern about petty thievery, I was concerned about getting mugged or assaulted. So my guess is that your chance of getting murdered is higher in Washigton DC than in Rome but I don't think you need to worry all that much about the pick pockets.