  Let X be a smooth complex projective variety of dimension d. It is classical
that ample line bundles on X satisfy many beautiful geometric, cohomological,
and numerical properties that render their behavior particularly tractable. By
contrast, examples due to Cutkosky and others have led to the common impression
that the linear series associated to non-ample divisors are in general mired in
pathology.
  However starting with fundamental work of Fujita, Nakayama, and Tsuji, it has
recently become apparent that arbitrary effective (or "big") divisors in fact
display a surprising number of properties analogous to those of ample line
bundles. The key is to study the properties in question from an asymptotic
perspective. At the same time, many interesting questions and open problems
remain.
  The purpose of the present expository note is to give an invitation to this
circle of ideas. In the hope that this informal overview might serve as a
jumping off point for the technical literature in the area, we sketch many
examples but provide no proofs. We focus on one particular invariant -- the
"volume" of a line bundle -- that measures the rate of growth of the number of
sections of powers of the bundle in question.
