The centro-affine geometry studies the subgroup of the projective group which leaves invariant (or interchanges) a point 0 (the center) and a plane not contaning 0. The research in this field was initiated by \textit{G. Tzitzeica} [Sur une nouvelle classe de surfaces, C. R. Acad. Sci., Paris 145, 132-133 (1907)]. The paper under review initiates the study of submanifolds in complex centro-affine spaces. The author's point of view is that this is a necessary step for attempting a study of submanifolds in complex affine spaces. The author introduces the so-called special hypersurfaces: they are characterized by the property that the centro-affine normal at any point belongs to the tangent hyperplane at this point. Adapting the definition of CR-submanifolds [\textit{A. Bejancu}, Geometry of CR- submanifolds (1986; Zbl 0605.53001)] the notion of a proper CR- hypersurface is defined and its equivalence with that of a special hypersurface is shown. On the other hand it is proved that special hypersurfaces are naturally endowed with a normal almost contact structure and a necessary and sufficient geometric condition for this to be pseudo-Sasakian is given. Finally, it is shown that nondegenerate hyperquadrics with center 0 are special iff they are invariant relative to the complex structure of \(C_{2n}\). The paper is written in a modern language, following the notations and terminology from \textit{P. A. Shirokov} and \textit{A. P. Shirokov} [Affine Differentialgeometrie (Teubner 1962); translation from the Russian original (Moscow 1959; Zbl 0085.367)].