Debuting as a pianist at the age of eleven, Kovacevich moved to London in 1958 to study with Myra Hess and had been living in the UK ever since. Stephen made a critically acclaimed British debut at Wigmore Hall in 1961, performing Sonata by Alban Berg, Preludes and Fugues by Johann Sebastian Bach, and Beethoven's Diabelli Variations. In 1967, Bishop gave his first concert in New York. He collaborated with Jacqueline Du Pré, Steven Isserlis, Nigel Kennedy, Lynn Harrell, Sarah Chang, Gautier Capuçon, Renaud Capuçon, Emmanuel Pahud and other prominent musicians, touring in Europe, Far East, Australia, New Zealand, South America, and the United States.
As a conductor, Stephen Kovacevich performed with the London Mozart Players, Royal Liverpool Philharmonic Orchestra, and Vancouver Symphony Orchestra.