The museum was established by the public foundation of the Marina Tsvetaeva Museum in June 1991, during the Tsvetaeva Poetry Festivals, now known as the Tsvetaeva Poetry Festivals, which have been held in the city of Alexandrov since 1982. It was the first museum dedicated to Tsvetaeva to open in Russia. That same year, the foundation transferred the museum to the administration of the city of Alexandrov, and it became municipal. The main principle of the museum is an individual tour for each visitor.

In 2008, the museum won the competition "The Changing Museum in a Changing World" with the project "Send Me a Garden for My Old Age."

The museum is conceived as a metaphor, recreating the atmosphere of the time in which the Tsvetaevas lived. The exhibition covers the years 1915 and 1917, when Anastasia and Marina Tsvetaeva resided in Alexandrov. The museum also features the exhibition "Alexandrov, the Capital of the 101st Kilometer," occupying two halls. A branch of the museum in the village of Lizunovo, the "House-Museum of Alexei Musatov," is dedicated to the life and work of the famous children's writer Alexei Musatov. The museum quarter includes two wooden houses, the "Tsvetaeva Cottage" and the "Lebedev House," a two-story stone house from the 16th and 20th centuries, as well as the Memorial Garden of Marina and Anastasia Tsvetaeva. The museum also includes the "Chamber Music Hall," which serves as both an exhibition hall and a manor in the village of Lizunovo. The exhibition "Alexandrov, the Capital of the 101st Kilometer" is located in a historic stone house from the 16th century on Military Lane, which was presumably used as the Tsar's stables of Alexandrov Sloboda.

The museum's collections number over thirty-five thousand items and include collections of materials from outstanding cultural figures whose activities are connected to the Alexandrov region. Among them are Marina and Anastasia Tsvetaeva, Osip Mandelstam, Ariadna Efron, Adelaida and Evgenia Gertsyk, Sergey Elpatyevsky, Alexander Solzhenitsyn, Anatoly Marchenko, Dr. Sergey Maslenikov, Hungarian writer József Lendvai, artist Lev Lagorio, Russian-Hungarian artist Viktor Toot, Swiss citizen Elsa Rutgers, priest Alexander Sergeenko, and many others.

The museum organizes three well-known festivals: the oldest poetry festival in Russia, the Tsvetaeva Poetry Festival, which has been held since 1982 in the last full week of June; the International Chamber Music Festival named after Sergey Korshunov, held since 1994 in the first full week of April; and the "Alexandrov Seasons of the Bolshoi Theater" festival. The annual Musatovsky Children's Literature Festival "Stozhary" is also held in the village of Lizunovo.