  The classical volume polynomial in algebraic geometry measures the degrees of
ample (and nef) divisors on a smooth projective variety. We introduce an
analogous volume polynomial for matroids, and give a complete combinatorial
formula. For a realizable matroid, we thus obtain an explicit formula for the
classical volume polynomial of the associated wonderful compactification. We
then introduce a new invariant called the volume of a matroid as a particular
specialization of its volume polynomial, and discuss its algebro-geometric and
combinatorial properties in connection to graded linear series on blow-ups of
projective spaces.

Combinatorics;Commutative Algebra;Algebraic Geometry