In the paper under review a non-trivial surjective endomorphism of a connected compact complex manifold \(X\) is understood as a surjective holomorphic map \(X\rightarrow X\) which is not biholomorphic. For dim \(X=1\) the Riemann-Hurwitz formula shows that \(X\) admits a non-trivial endomorphism if and only if it is isomorphic to the projective line. But in dimension two the determination of all \(X\) which admit non-trivial surjective endomorphisms is much more difficult. The projective algebraic case was solved by former results of both authors [see \textit{Y. Fujimoto}, Publ. Res. Inst. Math. Sci. 38, No. 1, 33--92 (2002; Zbl 1053.14049) and \textit{N. Nakayama}, Kyushu J. Math. 56, No. 2, 433--446 (2002; Zbl 1049.14029)]. The list consists of toric surfaces, \(\mathbb P^1\)-bundles over elliptic curves and specific \(\mathbb P^1\)-bundles over hyperbolic curves. In the present article they solve the non-algebraic case, thereby completing the two-dimensional list. The non-algebraic list consists of complex tori (algebraic dimension \(a(X)\leq 1\), \(b_1(X)\) even), the primary and secondary Kodaira surfaces and the elliptic Hopf surfaces (\(b_1(X)\) odd, \(a(X)=1\)), Hopf surfaces with two elliptic curves and specific Inoue surfaces of class \(VII_0\) (\(a(X)=0,b_1(X)=1, b_2(X)=0\)), and specific Inoue surfaces of class \(VII\). The authors prove that a compact complex surface \(X\) with a non-trivial surjective endomorphism contains only finitely many complex curves with negative self-intersection. This fact has strong implications. Case by case considerations and the use of the Enriques-Kodaira classification of compact complex surfaces yield the final list.