This timely collection of expository papers by leading researchers in the field highlights progress and new directions in Hopf algebras. The volume arises from the MSRI workshop on Hopf Algebras in October 1999, where many exciting recent developments were discussed. The work presented here will be stimulating to researchers and accessible to graduate students. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to recent work in quantum groups. In particular, there are chapters on recent progress in classifying finite-dimensional Hopf algebras, both in the semisimple case and in the pointed case, as well as what is known about the extension theory of Hopf algebras and on important connections of Hopf algebras to Lie algebras, knot theory and operator algebras. The volume also includes Mitsuhiro Takeuchi's article 'A short course on quantum matrices', now a standard reference in spite of its relative lack of availability; it has been updated for this volume.