I lived for 25 years in Rome. I had only one attempted break-in at home — they tried to force the front door, it may have been gypsies. I had some money stolen from me only once in all that time, on the Campidoglio steps. So, very safe, all round.
When I went to Lisbon, I had some money stolen because I was not cautious enough. So, just take the usual precautions, as you would in any large city. But overall, I would say that Rome is one of the safest cities to be in. The figures bear this out too.
Look at robbery rates in the USA. The biggest offender? Washington, D.C., whose 724 robberies per 100,000 people make it almost ten times more dangerous than Italy.
In general, Rome is safer than many other European capitals. I found that the biggest threat to my safety was the traffic and also the rather uneven pavements — you just have to take extra care in crossing the street. Respecting pedestrian crossings seems to be an optional! Scooters can be scary.
This is what Trip Advisor says:-
Probably the biggest safety risk in Rome is traffic. Be careful crossing streets in the congested city centre: it’s safer to cross with other people. Keep your eyes peeled for reckless scooters. Even if you cross at zebra crossings, you will find that some Romans will not slow down, although they will try to veer around you. If there is a pedestrian traffic light, you still need to look both ways to be sure it’s safe to cross. Be careful when crossing the road in squares as there tends to be little in the way of traffic control amd it appears to be a bit of a free for all. Piazza Venezia is a place to take particular care as traffic seems to come from all directions. Watch your step when walking out as street maintenance can leave a little to be desired. Pot holes, broken slabs and uneven paving are reasonably common.
As regards gun crimes, well USA is really dangerous (7 times more than Italy) and being murdered is much less likely in Italy (homicide in the USA is about 5 times more) while rape is 4 times more likely in the USA.
You can read all the crime stats for Italy compared to the USA here at Nation Master.