  The main goal of this paper is to study varieties with the best possible Mori
theoretic properties (measured by the existence of a certain decomposition of
the cone of effective divisors). We call such a variety a Mori Dream Space.
There turn out to be many examples, including quasi-smooth projective toric (or
more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds.
We characterize Mori dream spaces as GIT quotients of affine varieties by a
torus in a manner generalizing Cox's construction of toric varieties as
quotients of affine space. Via the quotient description, the chamber
decomposition of the cone of divisors in Mori theory is naturally identified
with the decomposition of the G-ample cone from geometric invariant theory. In
particular every rational contraction of a Mori dream space comes from GIT, and
all possible factorizations of a rational contraction can be read off from the
chamber decomposition.
