  The volume of a Cartier divisor is an asymptotic invariant, which measures
the rate of growth of sections of powers of the divisor. It extends to a
continuous, homogeneous, and log-concave function on the whole N\'eron--Severi
space, thus giving rise to a basic invariant of the underlying projective
variety. Analogously, one can also define the volume function of a possibly
non-complete multigraded linear series. In this paper we will address the
question of characterizing the class of functions arising on the one hand as
volume functions of multigraded linear series and on the other hand as volume
functions of projective varieties. In the multigraded setting, relying on the
work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that
any continuous, homogeneous, and log-concave function appears as the volume
function of a multigraded linear series. By contrast we show that there exists
countably many functions which arise as the volume functions of projective
varieties. We end the paper with an example, where the volume function of a
projective variety is given by a transcendental formula, emphasizing the
complicated nature of the volume in the classical case.
