  We investigate the Cox ring of a normal complete variety X with algebraic
torus action. Our first results relate the Cox ring of X to that of a maximal
geometric quotient of X. As a consequence, we obtain a complete description of
the Cox ring in terms of generators and relations for varieties with torus
action of complexity one. Moreover, we provide a combinatorial approach to the
Cox ring using the language of polyhedral divisors. Applied to smooth
k*-surfaces, our results give a description of the Cox ring in terms of
Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all
Gorenstein del Pezzo k*-surfaces with Picard number at most two and the Cox
rings of projectivizations of rank two vector bundles as well as cotangent
bundles over toric varieties in terms of Klyachko's description.
