On Axiom A Diffeomorphisms

These notes survey the results about Axiom A diffeomorphisms obtained since Smale's well-known paper of 1967. In that paper Smale defined these diffeomorphisms and set up a program for dynamical systems centered around them. These examples are charming in that they display complicated behavior but are still intelligible. This means that there are many theorems and yet some open problems. The first two sections copy Smale's paper for completeness. The last sections deal with certain non Axiom A systems that have received a good deal of attention. These systems display a certain amount of Axiom A type of behavior.