DON’T be surprised if you’re stopped by the police
Although new laws were passed in 2011 that ostensibly restructured Russia's police and their interactions with the public, it's still a good idea to carry a photocopy of your passport, visa, and registration – as well as travel documents that indicate how and when you'll return home – and present them to an officer if they demand to see them. Particular tourist police, with special training and language abilities, may be seen near important attractions like the Red Square to assist visitors.
If you're fined, Russian officials may anticipate a "unofficial payment" to speed up their service on the spot rather than dealing with the case later at the station. In any event, always request a formal receipt and keep the phone number for your country's embassy handy in case things get problematic. This one of the Things about Russia You Should Know before visiting Russia.