Zeltser was born in Kishinev in 1947. His mother Bertha was a concert pianist and teacher.Her grandfather, Mark Pester, was a well-known Bessarabian violinist (a classmate of Jascha Heifetz, Mischa Elman and Efrem Zimbalist at the St. Petersburg Conservatory) and conductor. Mark Zeltser was a child prodigy in mathematics. He began his piano studies with his mother. At eight he gave his first public recital and at nine performed concertos by Haydn and Grieg. He was admitted to the Moscow Conservatory without being required to take the entrance examinations due to his proven talents. It was the only time that such a concession has ever been made in the history of the Moscow Conservatory. 　　While still a student, Mark Zeltser entered and won several major competitions, including the National Competition in Moscow, Grand Prix at the Marguerite Long-Jacques Thibaud Competition in Paris, and the Busoni Prize at the Ferruccio Busoni International Piano Competition in Bolzano, Italy. He graduated from the conservatory with a Doctor of Musical Arts degree. Zeltser then served as a professor at the Conservatory and at the Moldavian State Conservatory, before beginning his performing career.