Museums: Historical
The Moscow Kremlin Museums.
http://www.kreml.ru/en-Us/museums-moscow-kremlin/
The State Historical Museum
Museums: Architectural
The Schusev Museum of Architecture
Museums: Fine Art
The Pushkin State Museum of Fine Arts
http://www.arts-museum.ru/?lang=en
The Tretyakov Gallery
https://tretyakovgallery.ru/en/
The Museum of Russian Impressionism
Museums: Theatre
The Bakhrushin Theatre Museum
http://www.gctm.ru/en/home-en/
Modern Art:
The Moscow Museum of Modern Art
Garage Museum of Contemporary Art
The Multimidia Art Museum Moscow
ArtPlay Design Center
Photographic Centres:
The Lumiere Brothers Centre for Photography
Museums: Jewish History:
Jewish Museum & Tolerance Centre
http://www.jewish-museum.ru/en/
Galleries:
V-A-C Foundation (It is dedicated to the development and international presentation of Russian contemporary culture)
http://www.v-a-c.ru/foundation/
In Artibus Foundation
Stella Art Foundation
http://en.safmuseum.org/index.php
Regina Gallery
NB Gallery (Soviet Art)
A favourite haunt amongst foreigners in Moscow looking for Soviet period art, in which the gallery specialises. If you are buying, be careful to check that you are buying the work of an artist who is at least fairly well known.
Contemporary Artists