  In this expository note we discuss a class of graded algebras named Cox
rings, which are naturally associated to algebraic varieties generalizing the
homogeneous coordinate rings of projective spaces. Whenever the Cox ring is
finitely generated, the variety admits a quotient presentation by a quasitorus,
which resembles the quotient construction of the projective space. We discuss
the problem of the finite generation of Cox rings from a geometric perspective
and provide examples of both the finitely and non-finitely generated cases.
