Improvisation, as an educational tool for actors, was pioneered by Jacques Copeau and furthered by such teachers as Jacques Lecoq, Carlo Mazzone-Clementi, Viola Spolin and Keith Johnstone. Its great usefulness is in teaching the student to release, recognize and channel spontaneity.
It is also useful as a readily available platform for teaching a wide range of principles and skills.
Beyond the classroom, it lends freshness and the excitement of unpredictability to such genres as Commedia dell' Arte and Theatre Sport.
An actor who is comfortable with improvisation has an in-built safety-net should things go "wrong" on stage.
Improvisation that is loosely tied to the goals of a story is often used in film making, a medium where it only has to work once.
Whole genres of Jazz and Contemporary Dance have developed around improvisational techniques.
Even Western Classical Music has its traditions of improvisation; especially the church organ where it is often necessary to fill in and cover delays and transitions in the service. People used to travel long distances to hear Bach improvise on the organ, and even today there are concert organists who will accept being given a theme to improvise on before the public.
One could say that all acting should be improvisation. When presented with an elaborate Shakespearean text, for instance, one should learn the words so well that one can forget them. Then one goes on stage and improvises within the situation, responds to the circumstances, and because one is basically lazy, the words that pop into one's mouth, conveniently, are Shakespeare's!