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).