Thefts are not uncommon in Peru - particularly in the cities, and during local festivals, when crowds make it hard to keep an eye on bags and pockets. Remember: around half of the population lives below the poverty line, so don't walk around carelessly displaying valuables.
Scams include someone bumping into you, fighting kids or someone spilling something on your clothes - all to provide a distraction for a pickpocket.
Avoid camping alone in remote areas.
Public transport can be dangerous, especially minibuses which are often overfilled and can drive recklessly fast along mountain switchbacks.
Do not enter the poorer districts of Lima, they are not safe for tourists.
Tourism police ("Poltur") are employed in the main cities and towns. Thanks to Peru's dependence on tourism, they are pretty efficient.