This collection of expository papers highlights progress and new directions in Hopf algebras.

Most of the contributors were participants in the Hopf Algebras Workshop held at MSRI in late 1999, although the papers are not necessarily tied to lectures given at the workshop. The workshop was very timely, as much progress has been made recently within Hopf algebras itself (for example, some long-standing conjectures of Kaplansky have been solved) as well as in studying Hopf algebras that have arisen in other areas, such as mathematical physics and topology. The first two papers discuss progress on classifying certain classes of Hopf algebras.

In the paper by Andruskiewitsch and Schneider, pointed Hopf algebras are studied in terms of their infinitesimal braiding. Important examples of pointed Hopf algebras are group algebras and the quantum groups coming from Lie theory, that is, Uq(g), g a semisimple Lie algebra, introduced by Drinfel'd and Jimbo, and Lusztig's finite-dimensional Frobenius kernels u(g) where q is a root of unity. The classification of all finite-dimensional pointed Hopf algebras with abelian group of group-like elements in characteristic zero seems to be in reach today. In this classification, Hopf algebras that are closely related to the Frobenius kernels play the main role.

In the second paper, Gelaki gives a survey on what is known on finitedimensional triangular Hopf algebras. In a sense these Hopf algebras are close to group algebras. Over the complex numbers, triangular semisimple, and more generally triangular Hopf algebras such that the tensor product of simple representations is semisimple, are Drinfeld twists of group algebras of finite groups or supergroups.