  This paper provides a rigorous account on the geometry of forms on
supermanifolds, with a focus on its algebraic-geometric aspects. First, we
introduce the de Rham complex of differential forms and we compute its
cohomology. We then discuss three intrinsic definitions of the Berezinian sheaf
of a supermanifold - as a quotient sheaf, via cohomology of the super Koszul
complex or via cohomology of the total de Rham complex. Further, we study the
properties of the Berezinian sheaf, showing in particular that it defines a
right $\mathcal{D}$-module. Then we introduce integral forms and their complex
and we compute their cohomology, by providing a suitable Poincar\'e lemma. We
show that the complex of differential forms and integral forms are
quasi-isomorphic and their cohomology computes the de Rham cohomology of the
reduced space of the supermanifold. The notion of Berezin integral is then
introduced and put to the good use to prove the superanalog of Stokes' theorem
and Poincar\'e duality, which relates differential and integral forms on
supermanifolds. Finally, a different point of view is discussed by introducing
the total tangent supermanifold and (integrable) pseudoforms in a new way. In
this context, it is shown that a particular class of integrable pseudoforms
having a distributional dependence supported at a point on the fibers are
isomorphic to integral forms. Within the general overview several new proofs of
results are scattered.
