In this quite long and very well structured article the authors introduce the notion of combinatorial hypersurfaces, which are codimension 2 submanifolds of $\bbfC \bbfP^n$ invariant under complex conjugation whose real parts are codimension 1 submanifolds of $\bbfR \bbfP^n$. This concept appears after removing the convexity condition imposed by Viro to the lattice subdivisions of the Newton polytope, to construct real algebraic varieties with prescribed topology [see e.g. {\it O. Ya. Viro}, Russ. Math. Surv. 41, No. 3, 55-82 (1986)].\par The authors show that combinatorial hypersurfaces obey almost all known topological restrictions satisfied by real algebraic surfaces; among them let us quote that they satisfy the generalized Harnack inequality, the Gudkov-Rokhlin and the Gudkov-Krahnov-Kharlamov congruences, some kind of Comessati inequalities for combinatorial hypersurfaces in $\bbfC \bbfP^3$, and that those of degree $d$ in $\bbfC \bbfP^3$ are homeomorphic to nonsingular algebraic surfaces in $\bbfC \bbfP^3$ of the same degree.\par The paper can be viewed as the first step trying to answer the following questions:\par (i) How far are are combinatorial hypersurfaces from the algebraic ones?\par (ii) What are the main differences between the combinatorial hypersurfaces and the notion of flexible curve introduced by {\it O. Ya. Viro} [in: Topology, general and algebraic topology, and applications. Proc. Int. Conf.,Leningrad 1982, Lect. Notes Math. 1060, 187-200 (1984)]?\par It must be pointed out that, as the authors recognize, the notion of combinatorial hypersurface was firstly introduced, with an slightly different language, in the pioneer work of {\it F. Santos} [``Improved counterexamples to the Ragsdale conjecture'', Univ. de Cantabria, Spain, Preprint 1994].
