Kathmandu has a rich and vivid art community and important art museums:

Museum of Nepali Art — MoNA
Traditional and contemporary Nepali art: paintings, sculpture, religious art, and modern works. It presents Nepali art as a living continuum, not only as heritage.

National Museum of Nepal (National museum with art collections)
Broad cultural-historical collection: weapons, antiquities, wood, bronze, stone works, paintings, and historical objects. It is not only an art museum, but it contains important art and craft material. 

Nepal Art Council (Exhibition institution / gallery)
Promotion of Nepali art and artists; hosts exhibitions rather than operating mainly as a permanent collection museum. 

Kathmandu Contemporary Arts Centre (KCAC) associated with Patan Museum area (Contemporary art centre / residency)
Contemporary art, international exchange, residencies, experimentation, and dialogue between Nepali and international artists. 

Patan Museum, Lalitpur/Patan, Kathmandu ValleyMajor art and heritage museum
One of the strongest museums for traditional Nepali sacred art, especially Hindu-Buddhist sculpture, metalwork, ritual objects, and palace-court culture. It is in Patan, not Kathmandu city, but it is essential for Kathmandu Valley art.

National Art Gallery, Bhaktapur, Kathmandu ValleyArt gallery / museum
​Traditional Nepali painting and historical art in a palace setting; important if you include the wider valley rather than only Kathmandu city.