The biggest risks to your safety in Europe like in any major tourist area are pickpockets and muggings. Using common sense and being aware of your surroundings can help to greatly reduce the risk of these occurrences. Remember alcohol is an integral part of many European cultures but overuse can lead to violence and poor judgement! In general, bars and pubs are not a place where alcohol causes these problems in Europe but it can end up being a big problem on the roads.
Most European countries have very low levels of violence compared to the United States. The main issues are drug use and gang related violence which are most prone in Britain and France, but it's virtually unheard of for any tourists to be involved in such issues. The few "trouble areas" to avoid are the run-down suburbs of certain urban areas (particularly in Europe's largest cities); some places in eastern and southern Europe do have much higher violent crime rates and can be very dangerous for non locals, but these areas shouldn't be of interest to the average tourist. Central and Western Europe are generally the safest regions.
Europe may be very urban and densely populated in general but as always when traveling in rural and forested / mountainous areas take the proper precautions. All it takes is one wrong turn down a ski piste and you are stranded. Time to take out the cell phone. Did you bring one?
For more information see Common scams which contains many Europe-specific scams.