Russia's obsession for foreign currencies, especially the U.S. dollar and more recently the Euro, supports a multitude of cash exchange venues throughout the city.
These places are quite small (and safe) often housing several numbered partitioned exchange windows.
The advertised exchange rate is not necessarily the one you receive as it may apply to transactions over a certain amount (often $500 or more), and some places, will and do discriminate against foreigners.
An armed uniformed security person is generally in attendance to organise the proceedings, ensuring only one person approaches a booth at any one time.
An original passport (not a copy) is required for the majority of exchange offices. You will normally receive details of the transaction in the form of a green printed receipt. This paperwork is purely for your records, and is not required when leaving Russia.
Larger more established exchange offices and banks accept travellers' cheques.
Tip:Various money exchange offices attract "Independent dealers" touting their own rate. The chances are they are profession con-artists that have an array of tricks to extract cash from you. Secondly, the saving if legitimate is very minimal. Avoid them.