Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely: โข control theory โข classical mechanics โข Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) โข diffusion on manifolds โข analysis of hypoelliptic operators โข Cauchy-Riemann (or CR) geometry. Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists: โข Andrรฉ Bellaรฏche: The tangent space in sub-Riemannian geometry โข Mikhael Gromov: Carnot-Carathรฉodory spaces seen from within โข Richard Montgomery: Survey of singular geodesics โข Hรฉctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers โข Jean-Michel Coron: Stabilization of controllable systems.