  We introduce and study Hodge-de Rham numbers for compact almost complex
4-manifolds, generalizing the Hodge numbers of a complex surface. The main
properties of these numbers in the case of complex surfaces are extended to
this more general setting, and it is shown that all Hodge-de Rham numbers for
compact almost complex 4-manifolds are determined by the cohomology, except for
one (the irregularity). Finally, these numbers are shown to prohibit the
existence of complex structures on certain manifolds, without reference to the
classification of surfaces.
