Trains arrive in Moscow at Kursk Station from central Russia (Kursk and Nizhny Novgorod), eastern Ukraine (Donetsk and Kharkiv), and from St. Petersburg (some Sapsan high-speed services). It is roughly 4km from the station to the Assambleya Nikitskaya Hotel.
By taxi
It should not take more than 20 minutes to reach the hotel by car in reasonable traffic conditions (although these are often not guaranteed in Moscow). Taxis should cost around $20, but if you think you will have trouble negotiating with a driver on arrival, it is probably safer to book a transfer, which you can do here.
By metro
Take the dark-blue line from Kurskaya Metro Station (attached to the mainline station concourse) and travel two stops west to Arbatskaya, which is around five minutes' walk from the Assambleya Nikitskaya Hotel.
