Born in Moscow, Boris Sveshnikov enrolled in the Moscow Institute of Applied and Decorative Arts in 1944. One day in 1946, at the age of 19, he left his parents' home to buy art supplies and was arrested by the Soviet Secret Police. He served eight years in prison and in a labor camp. During this 'camp' period he secretly produced hundreds of drawings and dozens of paintings while working as the camp's night watchman. Many of these works survive only because they were smuggled out of the camp. Sveshnikov depicted many images from camp life but was inspired by his earlier life, as well. He was influenced by the German and Dutch masters, and he also painted in a Surrealist style, although he had not been formally exposed to the movement. Following his release from the camp in 1954, he moved to the town of Tarusa, since former prisoners were required to live over 101 kilometers from major cities. In 1956, he returned to Moscow, to his parents' home. In his 'Moscow' period, the thirty years following his release from the camp, Svenshikov continued to be influenced, in part, by his camp years, and used a wide range of techniques to express what he saw in his mind's eye.

In 1956, Sveshnikov joined the Union of Soviet Artists, although he never participated in official exhibitions nor painted any State-commissioned works. For years, his work was shown in exhibitions of unofficial Soviet art in the United States, Europe and Japan, but never in the USSR. Exhibits include the World Exhibition Expo-58 (Brussels, 1959); The New School of Moscow (Galerie Interior, Frankfurt, and Galerie Behr, Stuttgart, 1969); Seven from Moscow (Galerie Christian Brandstatter, Vienna, 1975); New Soviet Art (Venice Biennale, 1977); New Art from the Soviet Union (Johnson Museum of Art, Cornell University, Ithaca, New York, 1977); Years of Soviet Unofficial Art 1956-1981, (Museum of Contemporary Russian Art, Montgeron, France, 1981-1982; and Boris Sveshnikov (solo exhibition at the Mimi Ferzt Gallery, New York, 1999).