The content of this paper is a classification of all two dimensional complex manifolds which are homogeneous under the action of a real Lie group of holomorphic automorphisms. The problem of classifying homogeneous surfaces has been considered in various different settings and of these results we note the following. Compact homogeneous surfaces were classified by \textit{J. Tits} [Comment. Math. Helv. 37, 111-120 (1962; Zbl 0108.363)], all complex homogeneous surfaces (a complex Lie group acts holomorphically) were given by \textit{A. T. Huckleberry} and \textit{E. L. Livorni} [Can. J. Math. 33, 1097-1110 (1981; Zbl 0504.32025)] and the list of pseudo-concave homogeneous surfaces is contained in the work of \textit{A. T. Huckleberry} and \textit{D. Snow} [Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV. Ser. 8, 231-255 (1981; Zbl 0464.32019)]. Assuming \(X=G/H\), where X is neither complex-homogeneous nor a product of homogeneous Riemann surfaces, we use the \({\mathfrak G}\)-anticanonical fibration of \textit{A. Huckleberry} and \textit{E. Oeljeklaus} [''Classification theorems for almost homogeneous spaces'', Inst. Elie Cartan, Univ. Nancy I 9 (1984; Zbl 0549.32024)] to show that X is a covering of \(Y:=G/I,\) which is an open submanifold of \(\hat Y:=\hat G/\hat I,\) a complex-homogeneous surface equivariant (but not necessarily closed) in some \({\mathbb{P}}_ N\). Further the algebraic group \(\hat G'\) acts transitively on \(\hat Y,\) so \(\hat Y\) is Zariski open in its Zariski closure. If no solvable group acts transitively on X, then X is either pseudoconcave and the group acting is semisimple, or it is \({\mathbb{C}}^ 2\setminus {\mathbb{R}}^ 2\) or one of its coverings and the group is a semi- direct product of \({\mathbb{R}}^ 2\) with \(SL_ 2({\mathbb{R}}^ 2)\). If the group acting is solvable, then \(\hat Y={\mathbb{C}}^ 2\) and the problem is to analyse fibrations G/I\(\to G/J\) with one dimensional fiber and base. The fact that these are real analytic but not holomorphic bundles makes this situation delicate. The boundary of G/I in \(\hat G/\hat I\) is studied and a key point in one of the cases is the uniqueness of the left-invariant CR-structure on the three-dimensional Heisenberg group [\textit{R. Tolimieri}, Trans. Am. Math. Soc. 239, 293-319 (1978; Zbl 0398.22017)] together with an extension theorem of \textit{N. Tanaka} [J. Math. Soc. Japan 14, 397-429 (1962; Zbl 0113.163)] in order to show that Y is either the ball or the complement in \({\mathbb{P}}_ 2\) of the union of the closed ball with a line tangent to it. Again the possibility \({\mathbb{C}}^ 2\setminus {\mathbb{R}}^ 2\) but with a solvable group acting occurs.