In recent years, the field of differential geometric control theory has grown and expanded in many directions. A particularly fruitful line of development has been that of putting the new tools to use to attack old control theory problems such as the structure of optimal trajectories and optimal synthesis, local and global controllability, system invertibility, sampling, canonical forms, and the structure of reachable sets. The purpose of this volume is to present an overview of recent results in this direction, and of the techniques used to derive them. It is based on the lectures given by the participants in a workshop on Finite Dimensional Controllability and Optimal Control held at Rutgers University on May 18 to 22, 1987. The workshop brought together a representative group of researchers in the field in order to produce an account of the current state of the art and explore directions of future work. The book contains, in addition to the chapters written by the workshop's participants, a paper kindly contributed by A. A. Agrachev and R. V. Gamkrelidze at the Editor's invitation.