Yes, I know several people that do. Just ask any taxi driver what parts of town are not safe. It would depend on the places you go. Every big city seems to have a "bad side of town." Just know where not to walk at night. Pickpockets and gypsis are the most common issues. If you're doing the tourist loop of Rome, London, Paris, Cannes ect you shouldn't have any major problems. However, certain regions like Naples have a reputation for crime. If you feel safe walking around Chicago, New York, Seattle ect with your kids then you should be fine in most major European cities. If by Mediterranean you mean Isreal, Turkey, Morroco ect. then I would use more caution. Check with the CIA's travel site for recommendations on how to stay safe and places to avoid. for sure I would bring my kids to Australia. Again, I'm sure there is one part fo town you shouldn't walk down alone with a small child. For most western countries, the same rules you use to stay safe at home still apply, but with a tad extra bit of awareness.