In this paper the notion of
∗
-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the
∗
-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of
∗
-Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing
∗
-Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose
∗
-Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.
Non-Hopf Hypersurfaces in 2-dimensional Complex Space Forms