At a recital last month in Seoul, the pianist and musicologist Robert Levin began the program's second half by pulling four slips of paper out of a basket. Then he launched into a musical fantasy that, to a layman's ear, sounded just like Mozart. It was Mr. Levin's own spontaneous composition, invented on the spot using suggestions gathered from the audience.

The art of improvisation, long dormant in classical music, is undergoing a revival in concert halls, conservatories and recording studios. A handful of performers say...