Try to fly into the renovated Domodedovo airport if you decide to come to Moscow. It isn't necessarily any faster-moving than Sheremetevo, but it is a precinct of the new Russia rather than the old: it's modern, relatively user-friendly and - most importantly - it allows you to escape extortionate taxis for the ride into town. There is an air train available there just outside the main terminal that costs just 200 rubles (as of May, anyway) and whisks you to Paveletskii station in about 45 minutes. There you can connect to the metro and anywhere else you want to go.
Domodedovo and Sheremetyevo airports are both outrageous for the time and cost of getting into the centre of the city. But both have minibuses travelling to near the end of the metro (Domodedovskaya or Rechnoi Vokzal on the green line) for 50 roubles. From there, you can catch the metro anywhere for 13 roubles. Domodedovo also has a train like the Heathrow Express, but it only goes every hour and costs 120 roubles, but then you need either the metro or a taxi from Paveletskaya.
A taxi from both airports will cost around 1300 roubles, but can take from 90 minutes to three hours, depending upon traffic.
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