Browning was born to musical parents in Denver, Colorado, in 1933. He studied piano from age 5 with his mother and, at the age of 10, was accepted as a student by Rosina Lhévinne. He appeared as a soloist with the Denver Symphony Orchestra later that same year.

In 1945 his family moved to Los Angeles, California. He spent two years at Occidental College there. He began his studies at the Juilliard School in New York with Rosina Lhévinne in 1950. He won the Leventritt Competition in 1955 and made his professional orchestral debut with the New York Philharmonic in 1956. At this point his career came under the management of well known talent manager Herbert Barrett with whom he remained for the rest of his career.

In 1962 he gave the premiere of Samuel Barber’s Pulitzer Prize-winning Piano Concerto, which was written for him, in connection with the opening of Lincoln Center. His second recording of the work, with Leonard Slatkin and the Saint Louis Symphony Orchestra in 1991 for RCA Victor, won a Grammy Award for best instrumental soloist with orchestra. Browning won a second Grammy in 1993 with a disc of Barber’s solo works on MusicMasters. He continued to follow the works of contemporary American composers but found relatively few to his liking.

Browning developed a busy career, giving some 100 concerts a season. He eased his schedule in the 1970s, explaining later that he had grown ragged from overwork. In the 1990s, his career had something of a renaissance. His last public appearance was at the National Gallery of Art in Washington in April 2002.

His last performance of all was to an invited audience at the United States Supreme Court in May 2002. He died at the age of 69 some eight months later in Sister Bay, Wisconsin.