  A classification of normal affine surfaces admitting a $\bf C^*$-action was
given in the work of Bia{\l}ynicki-Birula, Fieseler and L. Kaup, Orlik and
Wagreich, Rynes and others. We provide a simple alternative description of such
surfaces in terms of their graded rings as well as by defining equations. This
is based on a generalization of the Dolgachev-Pinkham-Demazure construction in
the case of a hyperbolic grading. As an apllication we determine the structure
of singularities, of the orbits and the divisor class groups for such surfaces.
