His parents were from Italy. His paternal grandfather Caesar Pugni is a famous ballet composer. Father Albert (in Orthodoxy Andrei) Puni is a cellist of the Mariinsky Theater.
In the years 1915−1916, Ivan Puni was a member of the creative union of avant-garde artists "Supremus", the creator of which was Kazimir Malevich.
In 1919, Ivan Puni taught at the Art Institute in Vitebsk under the supervision of Marc Chagall.
Ivan Puni was married to Xenia Boguslavskaya, who was also an avant-garde artist, as well as a designer and poetess. In the early 1920s, they emigrated — first to Germany, then to France.
In 1946, Ivan Puni took French citizenship.