We introduce the wedge product of two polytopes. The wedge product is
described in terms of inequality systems, in terms of vertex coordinates as
well as purely combinatorially, from the corresponding data of its
constituents. The wedge product construction can be described as an iterated
``subdirect product'' as introduced by McMullen (1976); it is dual to the
``wreath product'' construction of Joswig and Lutz (2005).

One particular instance of the wedge product construction turns out to be
especially interesting: The wedge products of polygons with simplices contain
certain combinatorially regular polyhedral surfaces as subcomplexes. These
generalize known classes of surfaces ``of unusually large genus'' that first
appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and
Wills (1983). Via ``projections of deformed wedge products'' we obtain
realizations of some of the surfaces in the boundary complexes of 4-polytopes,
and thus in R^3. As additional benefits our construction also yields polyhedral
subdivisions for the interior and the exterior, as well as a great number of
local deformations (``moduli'') for the surfaces in R^3. In order to prove that
there are many moduli, we introduce the concept of ``affine support sets'' in
simple polytopes. Finally, we explain how duality theory for 4-dimensional
polytopes can be exploited in order to also realize combinatorially dual
surfaces in R^3 via dual 4-polytopes.