  We give a classification of normal affine surfaces admitting an algebraic
group action with an open orbit. In particular an explicit algebraic
description of the affine coordinate rings and the defining equations of such
varieties is given. By our methods we recover many known results, e.g. the
classification of normal affine surfaces with a `big' open orbit of Gizatullin
and Popov or some of the classification results of Danilov-Gizatullin, Bertin
and others.
