While planning your trip to Russia, please check the information on what documents you need for staying in Russia.
To stay in Moscow, you must have following documents:

Part A of the migration card should be given to the passport control officer on the state border upon the entrance to the territory of the Russian Federation.
Part B of the migration card should be maintained for the whole duration of stay in the territory of the Russian Federation, and should be given to the passport control officer on the state border upon the exit from the territory of Russia.

The host party (in our case - the Institute) should notify the Federal Migration Service (FMS) of the Russian Federation on the arrival of a foreign citizen. The foreign citizens take the detachable part of the Notification (see photo) and should have it with them anywhere they go. The fact that the foreign citizen has the detachable part of the Notification with the mark of reception, confirms his/her migration registration, and thus it complies with immigration legislation of the Russian Federation.
To see the official information on the migration legislation in Russia, click here.