In recent years there has been enormous activity in the theory of
algebraic curves. Many long-standing problems have been solved using the
general techniques developed in algebraic geometry during the 1950's and
1960's. Additionally, unexpected and deep connections between algebraic
curves and differential equations have been uncovered, and these in turn
shed light on other classical problems in curve theory. It seems fair to
say that the theory of algebraic curves looks completely different now
from how it appeared 15 years ago; in particular, our current state of
knowledge repre- sents a significant advance beyond the legacy left by
the classical geometers such as Noether, Castelnuovo, Enriques, and
Severi. These books give a presentation of one of the central areas of
this recent activity; namely, the study of linear series on both a fixed
curve (Volume I) and on a variable curve (Volume II). Our goal is to
give a comprehensive and self-contained account of the extrinsic
geometry of algebraic curves, which in our opinion constitutes the main
geometric core of the recent advances in curve theory. Along the way we
shall, of course, discuss appli- cations of the theory of linear series
to a number of classical topics (e.g., the geometry of the Riemann theta
divisor) as well as to some of the current research (e.g., the Kodaira
dimension of the moduli space of curves).