I've come across a number of scams while out and about in my travels. The gypsy girls in Paris that ask, Do you speak English? to get you to sign a "petition" in French that says you will give them money? Yup, saw that a number of times in my recent trip to Paris. I simply ignored them, didn't make eye contact, and walked in the other direction.
In Florence, there will be jewelry sellers that try to give you a bracelet "regalo" (gift) and then demand money. One time someone tried doing that to me, even though I didn't make eye contact or reply to them. He even went so far as to follow me into a cafe and kept trying to pester me while I continued to ignore him. Even though I never touched the bracelet, he tried demanding five euros. He finally got the hint and left.
Scams are inevitable, but the best defense is knowledge. Always do your research before heading somewhere new to be aware of what scams could be present. This handy infographic displays a number of popular scams to look out for: