He studied violin under Piotr Stolyarsky, Yuri Yankelevich, and Adolf Lechinsky (student of Carl Flesch), and composition under Aram Khachaturian.

[4] The New York Times wrote: “The audience roared approval, coming to its feet for three standing ovations, Mr. Markov wowed them with dazzling pyrotechnics.”[4] Following this, Markov appeared as soloist alongside orchestras and in solo recitals in Carnegie Hall, The Lincoln Center, and the Kennedy Center, and performed in the concert halls of Chicago, Los Angeles, Philadelphia, Houston, Toronto, Montreal and other cities of North and South America, Europe, Asia, and Africa.

[4] He has collaborated and/or recorded with pianists David Ashkenazi, Bella Davidovich, Oxana Yablonskaya, Andrey Mytnik, Robert Levin, Milton Kaye, and Dmitri Kogan.

Albert Markov's recordings can be found on Melodiya, Musical Heritage Society, Sunrise, and RMS, with works of different periods including his own compositions.

[4] Markov has written many compositions, including his "Chinese" violin concerto and his Formosa suite, which have been recorded with the Russian National Orchestra.

[6][circular reference] His wife Marina Markov is also a violinist who has played with the New York City Opera and the Bolshoi orchestra.