  The past decade has witnessed two important new developments in the study of
linear series on algebraic varieties. First, vector bundles have emerged as
powerful tools for analyzing linear series on curves and surfaces. More
recently, the viewpoints and techniques of higher dimensional geometry have
started to play a central role. The purpose of these lectures is to provide a
down-to-earth introduction to some of these new ideas and methods.
  The present notes, written with the assistance of G. Fernandez del Busto, are
a considerably expanded version of a course given at the 1993 Regional Geometry
Institute on Higher Dimensional Complex Geometry at Park City, Utah. While I
hope they may be of interest to any geometers wishing to learn about recent
work on linear series, I have particularly attempted to aim the discusion at a
novice audience. My hope is that with a little faith and effort, these notes
should be accessible to anyone having finished the standard introductory texts
in algebraic geometry. The underlying theme of the lectures is the search for
higher dimensional generalizations of the most basic facts about linear series
on curves. But in an attempt to keep things elementary we work more or less
exclusively on surfaces. Thus we end up discussing one central result -- viz.
Reider's theorem -- from many points of view. Sticking to the case of surfaces
allows one to eliminate many technical complexities, and I hope that parts of
these lectures might therefore serve as a useful first introduction to the
powerful cohomological techniques (involving $Q$-divisors) of contemporary
