Let \(M_ n (c)\) be an \(n\)-dimensional complete and simply connected complex space form, i.e. a complex Kähler manifold of constant holomorphic sectional curvature \(c\). Following results of \textit{J. Berndt} [J. Reine Angew. Math. 359, 132-141 (1989; Zbl 0655.53046)], \textit{M. Kimura} and \textit{S. Maeda} [Math. Z. 202, 299-311 (1989; Zbl 0661.53015)], \textit{R. Tagaki} [Osaka J. Math. 10, 495-506 (1973; Zbl 0274.53062)] and others, the authors consider a real hypersurface \(M\) of \(M_ n(c)\), \(c\neq 0\), \(n\geq 3\), satisfying certain sufficient conditions, and they show that \(M\) is locally a tube belonging to one of seven types, or \(M\) is a ruled real hypersurface. The well-organized proofs of the theorems are the content of this nice paper.