Russia is a famously criminal society, but most of that crime is restricted to wide-scale bribery and corruption. The high-profile killings and gang warfare that earned Moscow its gangster reputation in the early and mid-nineties are largely a thing of the past, and never affected tourists anyway.
Petty crime, as in all large cities, can be a problem in Moscow, and tourists are often the target - for the obvious reasons that they stand out in a crowd and are more likely to be carrying large amounts of cash or valuables. Pickpockets of various species are the main threat, and they tend to be most prevalent on public transport, especially the metro, and round tourist traps. To avoid being a victim, follow the obvious precautions: do not carry more cash than necessary and try not to display large sums in public places, keep large sums in a money belt or sealable inner pocket, keep credit cards separately, and don't carry valuables in a backpack or easily opened bag.
Violent crime of any sort is extremely unlikely to be a problem for tourists. Mugging is rare in Moscow, and you are only likely to encounter this late at night if you are doing something to be ashamed of. There are not really any 'no-go' areas of Moscow, or certainly none that a visitor is likely to wander into by mistake, but there is a constant low-level threat of drunken aggression from xenophobic thugs, so you are advised not to act too loudly or brashly in bars and clubs, and to be polite but firm with anyone who approaches you in the street and tries to strike up conversation.
The frustration for most foreigners with crime in Moscow is not the relatively low level, but the total inaction of the police. Moscow's Militsia are notorious for their corruption and inefficiency, and it often feels that you're lucky if your assailant is not being actively abetted by the nearest policeman.
If you are robbed while in Moscow, then for insurance purposes you will need to obtain a police report (the same goes if you lose anything valuable). It will be easiest to find the small police stations located in most metro stations and large hotels - few policemen speak English, so try to find someone Russian to help you.
The Militsia are also very keen on document checks and, although you aren't legally obliged to carry your documents with you, police are entitled to get confirmation of your identity by taking you into a police station. Most of the time they are on the look out to make some money, and may pretend to find fault in perfectly legitimate visas and registration cards. Patience and feigned (or genuine) incomprehension are the key here, as they are playing a fairly risky game and are unlikely to waste too much time with you. Whether you are better off carrying originals or photocopies of your documents is debatable - you risk losing the originals, but photocopies are likely to cause more trouble if you can't explain much in Russian. The phrase "moi dokumenti v OVIRe," - meaning that your passport is being registered - may come in useful.