Popular tourist sites like the Colosseum are hotspots for tourist-related crime.For a large city, Rome has a relatively low rate of violent crime. However, this is more than made up for by the high levels of pickpockets and various types of scammers that prey on the unwary, especially in high tourist areas like Termini station, the Vatican, and other major sites. You may want to familiarise yourself with the information on taxi scams and other tourist scams.
Hotspots
As is often the case, the criminals congregate where the tourists are: crowds of lost, tired, overheated tourists make for easy pickings. Be careful around the Vatican, and other crowded areas like the Spanish Steps and the Colosseum. Public transport is another danger zone: be cautious in main transport hubs like the airports and Termini station, and on busses, trams, and the metro (especially on crowded routes that move between tourist sites).
Pickpockets
Pickpockets are a big problem in Rome. When you're out wandering the streets or riding the metro, make sure that your wallet, phone and other valuables are stowed securely (and not in the back pocket of your jeans).
Pickpockets vary in appearance from gangs of itnerant gypsies to well-dressed individuals, and often carry something to cover their activities: a shopping bag, a map, or just a big piece of cardboard. There are also gangs of children who will mob you and, by the time you've shooed them away, your pockets have been systematically emptied.
Watch out for obvious distractions: if something's causing a scene and making a crowd gather, whether it's an impromptu musical performance, a street argument, or a woman taking her clothes off (it's been known), there's a good chance it's all been arranged so that a pickpocket can work the crowd.
Bagsnatchers on mopeds can also be a danger. If you must carry a bag, try to sling it over your head and across your body, with the bag hanging away from the road. (Some others advise the opposite: to carry the bag loose off one shoulder, so if it is grabbed by a passing motorino driver, you aren't dragged with it.)