Kremer was born in Riga to parents of German origin. He began to play the violin at the age of four, receiving tuition from his father and his grandfather, who were both professional violinists. He went on to study at the Riga School of Music and with David Oistrakh at the Moscow Conservatory. He won prizes at the Queen Elisabeth Competition in Brussels in 1967 (Second Prize), the Paganini Competition in Genoa in 1969 (First Prize) and the International Tchaikovsky Competition in Moscow in 1970 (First Prize).
Kremer's first concert in the West was in Germany in 1975, followed by appearances at the Salzburg Festival in 1976 and in New York in 1977. In 1981, Kremer founded a chamber music festival in Lockenhaus, Austria, with a focus on new and unconventional programming; since 1992 the festival has been known as 'Kremerata Musica' and in 1996 Kremer founded the Kremerata Baltica chamber orchestra, composed of young players from the Baltic region. He was also among the artistic directors of the festival 'Art Projekt 92' in Munich and is director of the Musiksommer Gstaad festival in Switzerland.
Kremer is broadly admired for his wide‐ranging repertoire, extending from Vivaldi and Bach to contemporary composers. He has championed the work of composers such as Ástor Piazzolla, George Enescu, Philip Glass, Alfred Schnittke, Lera Auerbach, Arvo Pärt, and John Adams. His partners in performance include Valery Afanassiev, Martha Argerich, Oleg Maisenberg and Vadim Sakharov. He has a large discography on the Deutsche Grammophon label, for which he has recorded since 1978, and he has also recorded for Philips and Decca. He plays a Guarneri del Gesù violin dating from 1730.