  In his work on log-concavity of multiplicities, Okounkov showed in passing
that one could associate a convex body to a linear series on a projective
variety, and then use convex geometry to study such linear systems. Although
Okounkov was essentially working in the classical setting of ample line
bundles, it turns out that the construction goes through for an arbitrary big
divisor. Moreover, this viewpoint renders transparent many basic facts about
asymptotic invariants of linear series, and opens the door to a number of
extensions. The purpose of this paper is to initiate a systematic development
of the theory, and to give a number of applications and examples.
