induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras and some surrounding topics. New proofs of the classical theorems are given, as well as some new results. The paper is divided into eight sections with the following titles: 1. Algebras and modules. 2. Coalgebras and comodules. 3. Algebras, coalgebras, and duality. 4. Elementary properties of Hopf algebras. 5. Universal algebras of Lie algebras. 6. Lie algebras and restricted Lie algebras. 7. Some classical theorems. 8. Morphisms of connected coalgebras into connected algebras. The first four sections are introductory in nature. Section 5 shows that, over a field of characteristic zero, the category of graded Lie algebras is isomorphic with the category of primitively generated Hopf algebras. In ? 6, a similar result is obtained in the case of characteristic p # 0, but with graded Lie algebras replaced by graded restricted Lie algebras. Section 7 studies conditions when a Hopf algebra with commutative multiplication splits either as a tensor product of algebras with a single generator or a tensor product of