There are thieves. Depends a lot on where you go. Don't forget about the possibility of losing your wallet while changing lenses, or framing a photo. Best if someone can watch your back in crowded places.
Crime against tourists in Europe is almost always the non-violent, crime-of-convenience sort of thing. You don't need to be paranoid, just more aware of your surroundings than when in familiar territory...
In all my travels in Europe I have never had a problem with crime because I take the simple precautions I list below. I have had my share of run-ins with gypsies but they never got close enough to get their hands on my stuff. I prefer taking public transportation whenever possible because it is less expensive, very convenient and a great way to meet people. But I carry as little as possible so I don't feel at risk riding in a crowded tram. A little information and common-sense goes a long way when traveling abroad.
1. Snatch-and-run/ride does occur, so purses, cameras and bags hanging from a shoulder (particularly with said bag dangling on the street side...) isn't a very good idea. I prefer the cross-body R Strap, a small bag like this
http://www.ebags.com/product/baggallini/messenger-slingbagg-crinkle-nylon/47360 or a daypack carried over both shoulders.
I use small locking carabiner fasteners on outside zippers. They can be found at outdoor supply stores. It takes several turns to open the lock. Remember that thieves want an easy target and not a confrontation.
2. Pickpockets are prevalent. Carry your passport, credit card, ATM card, etc. in a money belt. Only place an amount of cash that you are willing to lose in a pocket. Pockets with velcro closures are not a problem for an experienced pickpocket...
I once met an American tourist who lost his passport, ATM and credit cards and money because he thought that the velcro would deter pickpockets. A friend of mine lost over 400 Euro because in his weariness after a ten-hour plane ride he visited an ATM at the Amsterdam airport and placed the cash in an outside pocket of his pack. By the time he reached his hotel the money was Long Gone...
3. Thieves look for distracted tourists and most of the problems occur in tourist spots with big crowds. Crowded transportation hubs and buses/trams are also potential trouble spots. Don't stop in crowded areas or the middle of a sidewalk to look at a guidebook/map; this immediately pegs you as a distracted tourist. Get out of the flow of traffic, preferably against a wall if you and your companion need to focus on something at the same time.
If you need to use an ATM or get into your bag, have your companion be the "lookout" while you take care of business.
When on public transportation keep your bag on your lap or connected to your person in some manner. I carry a Pacsafe for securing my bag in locations when I cannot avoid being separated from it (which rarely happens.)
4. In cities like Paris and Rome gypsies are often involved in stealing from tourists. Don't allow them to get too close to you; you may have to be more assertive than normal. A woman carrying a child with older children in tow may try to distract you while the older children go for your valuables. Let suspicious people know that you are aware of them; a good way of doing this is to look right at them while pulling a lower eyelid down; this translates to
"I am keeping an eye on you!"
I feel much safer in strange cities in Europe than I do in urban areas in the U.S.. (Unlike U.S. cities some European suburbs can be more risky than the city centers. Paris Metro lines that go through certain suburbs are best avoided by tourists unless you are the adventurous type...) European countries tend to be generally much less violent, the private ownership of handguns is almost nonexistent and most of the time travelers are in the company of other people on the street. Once you get away from tourist areas the risk of theft drops off considerably, which is another good reason to stay off the beaten tourist track as much as possible.
The three most common mistakes I see American travelers do in Europe are:
1. Carrying too much stuff.
2. Trying to do too much.
3. Spending too much time viewing the world through a viewfinder or tiny LCD.
The more you have, the more you have to carry and keep track of. Expensive cameras, watches, etc. attract the kind of attention you don't want...
If you try to do too much during a trip you will usually wind up getting frustrated, overly-tired and you won't enjoy your trip as much. If you are tired you are more likely to let-down your guard and become a target for a pickpocket.
Traveling isn't about quantity, it's about quality. Instead of cramming-in visits to multiple sights in a day, pick one or two of your top choices while leaving sufficient time for eating, drinking and relaxing. Ask yourself who you are taking all those pictures for: yourself or to prove to family and friends that you actually visited Notre Dame? Try to stay in the moment and remember that the best memories are captured by the brain, not a still or video camera. You can replay them at will without worrying that the current digital media may change in the future, rendering your memories "unreadable"...
Know when to take the easy way out. In a city like Paris the time and trouble it takes to figure out public transportation can be considerable if you don't speak French. If you have the time and inclination, go for it... Otherwise, take advantage of the multitude of Mercedes taxis and be chauffeured to your destination. Sure, it is more expensive and probably slower than the Metro but you can relax and gaze out the window at the landmarks, something that is considerably more difficult to do underground...
As I previously suggested, a little reading before you leave can get you up to speed about what to see and what to avoid.