  For smooth manifolds equipped with various geometric structures, we construct
complexes that replace the de Rham complex in providing an alternative fine
resolution of the sheaf of locally constant functions. In case that the
geometric structure is that of a parabolic geometry, our complexes coincide
with the Bernstein-Gelfand-Gelfand complex associated with the trivial
representation. However, at least in the cases we discuss, our constructions
are relatively simple and avoid most of the machinery of parabolic geometry.
Moreover, our method extends to certain geometries beyond the parabolic realm.
