Improvisational theatre is as old as time. It pre-dates the invention of writing, since long before we started writing scripts we were telling stories by acting them out.
After the Commedia died off, improv theatre faded into obscurity until it was separately and spontaneously re-invented by two people who have shaped the craft as it exists today -- Keith Johnstone and Viola Spolin.
Johnstone decided that one approach would be to combine elements of both theatre and sports, to form a hybrid called Theatresports. The trappings of team sports were adapted to the improvisational theatre context; teams would compete for points awarded by judges, and audiences would be encouraged to cheer for good scenes and jeer the judges ("kill the umpire!").
Through Theatresports, Johnstone's ideas have gone on to influence (directly or indirectly) almost every major improv group.
Spolin's son, Paul Sills, built on his mother's work and was one of the driving forces of improvisational theatre centered around the University of Chicago in the mid-1950's. Along with people like Del Close and David Shepherd, Sills created an ensemble of actors who developed a kind of "modern Commedia" which would appeal to the average man in the street. As with Theatresports and the original Commedia, the goal was to create theatre that was accessible to everyone.
The group that sprang from the work of Sills, Shepherd and Close, called The Compass, was extremely successful. It brought people to the theatre who in many cases had never gone before, and eventually led to the development of a company called Second City.
Through The Compass and Second City, Spolin's Theatre Games have gone on to influence an entire generation of improvisational performers.