h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

There aren't any Taxis in Moscow. They were apparently declared illegal some years back. So, if you want to get around, you'll have to "catch a car" This means waving your hand out on any road to signal a car to pull over. Once this happens, haggle your price with the probably former taxi driver who was "just driving around", and you should be there direct. You'll need to know your numbers though!

I think there are official taxis in Moscow, and they are clearly recognisable... many of them bright (well... ) yellow machines type Wolga. Volkswagens, Mercedes are also seen as (official) taxi. I myself prefer Wolga, because I usually have a lot of luggage.
When arriving at Sheremetyevo airport you'll be surrounded by a horde of taxi drivers. Some of these may be official taxi drivers, most of them may be reasonably honest (if we consider charging 400% of the realistic rate honest). If you have a contact (company/hotel/..) I suggest you arrange airport transport with them. That costs about 1000~1500 roubles (30..50 Euro).
Negotiate a price before you get in, start with 25 to 30 % of the driver's suggestion. You'll end up paying 60 % of his first suggestion which by any international standard is too much.

Ideal is to have a Russian native speaker negotiate the price for a taxi.

The "catch a car" is possible.. the response of "private" cars goes up and down with the state of the local economy. You may be standing in the street (rain, snow, dust, mud) for a loooooong time.