  This is an expanded version of the author's 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.
