As you probably know, the city of Moscow itself is a separate subject of the Russian Federation and is not part of the region.
The offices of public authorities are partly located in Krasnogorsk, just outside the capital.
Most of the country houses, or dachas, of Muscovites are located in the suburbs. Residents of the capital also often come here for entertainment - for example, to natural resorts, spas or just for picnics on the banks of the Moscow River.
By the way, the Odintsovsky district of Moscow Region is considered to be the administrative center for the Baikonur cosmodrome. Yes, geographically it is part of Kazakhstan, but Russia has leased it until 2050!