Tourists should always carry their passports with them. Russian policemen often ask to see these as well as documentation proving that you registered yourself with the Russian authorities upon arrival in Moscow. Visitors should steer clear of policemen as some of them are extremely corrupt and always on the lookout for a bit of extra money. If you are stopped by them you can either pay them a bit of money, or if they hassle you too much it is best to threaten to call your embassy. It will usually not get to this though.
Do not walk alone late at night [this especially applies to women]. It is best to walk in groups and steer clear of drunken locals. Do not carry too much cash on you. Moscow, like most major metropolises, is swarming with pickpockets. Be careful while crossing roads- Russians are reckless drivers and stop for no one. In case of emergencies always carry the following numbers with you:
911 Rescue service: 937-9911
Lost children: 401-9011
Lost documents: 200-9957
Lost credit cards: 956-3556, 234-1831