While violent crime is still relatively low in Rome, there's a scammer or con-artist on pretty much every corner of the tourist areas. Below you'll find a list of the more common tricks. You may also want to read up on general crime in Rome, and on taxi scams.
There are plenty of scammers and thieves at large in Rome, but as long as you use common sense and are careful, you should be fine. Italians can be very friendly people, but that doesn't mean strangers in Rome are any more trustworthy than they are anywhere else.
If you think you're being targeted by a scammer, or feel in danger, don't panic and don't get violent (although he or she may seem to be alone, they almost certainly have friends close by). If possible, just walk away. If you're being cornered by somebody claiming to be a police officer, and you're not convinced of their credentials, suggest that you go together to speak with a uniformed officer (there are plenty around the historic centre). If you're feeling in immediate danger, just start shouting for the police. A few hearty bellows of "Polizia!" or "Carabinieri!" should clear your path.
Strangers Bearing Gifts
Beware of strangers as you would anywhere else. One scam involves a man in a car stopping you to ask for directions. He strikes up a conversation, tells you he workd in fashion, and presents you with an apparently very valuable item of clothing. Then he asks for a little money, as he's about to run out of petrol. The money he asks for is far more than the (fake) clothes are worth, and he'll turn nasty if he doesn't get it.
Bars and Strip Clubs
There have been a number of cases reported where young men have been approached by strangers who claim to be lost and looking for directions (a bit of a theme, this, as it's an easy way to start up a conversation with a stranger). They strike up a conversation and then suggest a drink. The bar they take you to is usually a strip club, where you will be presented with a bill for hundreds of euros, and it's made clear that deeply unpleasant things will happen to you if you don't pay up. More than a few young men have had their holidays cut short by this trick.
"We've Found Your Wallet"
One particularly elaborate scam kicks in once your wallet's been stolen, either by a pickpocket or by somebody cutting a hole in your bag. The scam consists of two calls: the first purports to be from the police, claiming they've found the wallet, and reported the cards as stolen to your bank. They give you a (false) address for a police station where you can reclaim your property. While you're wandering the streets looking for the phantom police station, a second call comes through, this one apparently from your bank, claiming that they're ready to put a stop on your card, but just need to confirm some security details. Stressed and tired, you confirm details, including your security PIN number...
The whole thing is just an elaborate way of getting your PIN number so that they can clean out your account. The worst part is that, because you've technically volunteered your number to the thieves, you may not even be covered by whatever fraud prevention your bank would usually offer. Remember, never give out your PIN number to anybody: not even your real bank will ever ask you for it.
"Tourist Police" / Checking for Counterfeit Money
This scam demonstrates just how easy it is to take advantage of confused tourists. You think "I'd never fall for this one", but so have a great many people who went on to be victims.
In this scam, you're approached by a man claiming to be from Rome's "Tourist Police". He flashes ID, and asks to check your documents, and the money in your wallet for counterfeit notes. Needless to say, when the wallet's returned to you and you're sent on your way, it's a little lighter.
A common variation on this scam involves a third party, a man who approaches you before the "tourist police" officer. He tries to speak to you in Italian (a neat way of checking how savvy you might be). Then the policeman arrives, and proceeds as above, with the neat pretext that you've just been seen "conversing with a known drug dealer". Here, your desire to avoid being arrested in a foreign country on drugs charges might override your normal sense when it comes to handing over that wallet.
Not a scam, exactly, but certainly a nuisance. These guys come up to you and start tying a friendship bracelet around your wrist. If you're too tired, hot, or distracted to chase them away, they'll try to charge you €20 or so for the bracelet—you can't just hand the thing back, though, because it's tied tightly onto your wrist. Tear it off, walk away, but don't get violent (he's probably not alone), and if he won't leave you, call for the police.