The cohomological properties of supermanifolds (intended in the sense of De Witt [Supermanifolds (Cambridge U. P., London, 1984)] and Rogers [J. Math. Phys. 21, 1352 (1980)]) are investigated, paying particular attention to the de Rham cohomology of supersmooth differential forms (SDR cohomology). The SDR cohomology of De Witt supermanifolds is shown to be equivalent to the de Rham cohomology of their body. The SDR cohomology is explicitly computed for some topologically nontrivial supermanifolds and some general conclusions concerning the geometric structure of supermanifolds and the properties of the SDR cohomology are drawn. In particular, it is shown that the SDR cohomology is neither a topological nor a real differentiable invariant, but rather a ‘‘superdifferentiable’’ invariant.