An introduction to tight closure

This is an expanded version of my lecture at the Conference in Commutative Algebra and Algebraic Geometry in Messina Italy in June 1999. The purpose of the talk was to give a brief introduction to the subject of tight closure, aimed at commutative algebraists who have not before studied this topic. The first part focused mainly on the definition and basic properties, with the second part focusing on some applications to algebraic geometry, particularly to global generation of adjoint linear series. These lecture notes follow even more closely a series of two lectures I gave in Kashikojima, Japan, at the Twentieth Annual Japanese Conference in Commutative Algebra the previous fall, and were distributed also in conjunction with that conference. I wish to thank the organizers of both conferences, Professors Restuccia and Herzog for the European conference, and Professors Hashimoto and Yoshida, for the Japanese conference. Both events were a smashing success. Special thanks are due also to Rosanna Utano, for help editing the tex file.