This definit ion is equivalent with the following one: A complex manifold V is called a lmost homogeneous if the group G opera tes transit ively outside a nowhere dense set S ~ I,(. If the set S, the singular set of I/, is void, V is called homogeneous. In this note we will prove the following Main Theorem. An almost homogeneous compact complex surface V is biregularly equivalent with a surface of one of the following types: (i) a rational surface; V is P2 (tI~) or V is obtained from a holomorphie Pl(IE)-bundle over PI(IE) by blowing up a .finite number of points, (ii) a topologically trivial Pl(tl~)-bundle over a one-dimensional torus, (iii) a HopJ" surface (i.e. a c o m p a c t complex surface of which the universal cover ing is 112 2 (0)) with abelian fimdamental group, (iv) a complex two-dimensional torus. The surfaces of type (ii), (iii) and (iv), the Pl(C)-bundles over PI(II~) and 1"2 (11~) are indeed a lmost homogeneous .