Honored Artist of the RSFSR (1969). People's Artist of the RSFSR (29 March 1977). People's Artist of the USSR (31 January 1985). His elder sister was the actress Rimma Markova. In 1931-1934 he played children's roles in productions of the Saratov Drama Theatre, where his father V. D. Markov worked as an actor. In 1945-1947 he studied at the studio of the Vologda Drama Theatre. He graduated from the studio at the Moscow Lenin Komsomol Theatre (1950). From 1950 he was an actor at the Moscow Lenin Komsomol Theatre. From 1960 he was an actor at the Pushkin Theatre. From 1965 he was an actor at the Academic Mossovet Theatre. In the 1986-1987 season he moved to the Maly Theatre. Leonid Markov possessed a unique voice and manner of speaking. That is probably why he was often invited to work on radio.

About a hundred recordings, covering a large period from 1954 to 1990, of literary works and radio plays featuring him are preserved in the radio fund collection: "In the Winter Forest" by S. Bogomazov, as the Wolf; "At the Gymnasium" by N. Garin-Mikhailovsky, as the Father; "White Roses, Pink Elephants" by W. Gibson, as Henry Pulaski; "Fate" by P. Proskurin, as Makashin; "Inspector Tikhonov" by Arkady and Georgy Vainer, as Sharapov; "Korolev and Gagarin" by V. Gubarev, as Korolev; "First Love" by I. Bunin, as the Father, and other productions. The actor read literary works by A. Pushkin, M. Lermontov, F. Dostoevsky, I. Abashidze, V. Astafyev, B. Vasilyev, R. Rozhdestvensky.