Kate, it's on a par with most large metropolitan cities such as LA, NYC, Paris or London. Walking alone at night is not recommended. If you are out late, get a taxi, but don't hail one off the street (assault of passengers is not rare), get the restaurant/bar to call one for you from the official service. Never use unofficial taxis.
Around the tourist haunts like Red Square during the day is generally fine. The street food is best avoided, there are NO hygiene rules for them. Drink spiking is sadly quite common in clubs. As you see, the usual big city precautions are required.
Phil from the safety hub over 8 years ago