He was appointed Young Conductor in Association to the Northern Sinfonia at the age of nineteen. In 1997 he became Principal Conductor of the London Philharmonic Youth Orchestra and two years later was invited by Seiji Ozawa to join the Boston Symphony Orchestra as Assistant Conductor. He was Chief Conductor of the BBC Scottish Symphony Orchestra from 2003 to September 2009, subsequently becoming Principal Guest Conductor. In January 2011, the Iceland Symphony Orchestra named Volkov as its 9th chief conductor and music director. With jazz musician Assif Tsahar he is also one of the guiding forces behind Levontin 7, a performance venue in Tel Aviv that brings together differing musical genres
He has a daughter with his partner, pianist/singer/composer Maya Dunietz.