  We provide a draft of a theory of geometric integration of rough differential
forms which are generalizations of classical (smooth) differential forms to
similar objects with very low regularity, for instance, involving H\"older
continuous functions that may be nowhere differentiable. Borrowing ideas from
the theory of rough paths, we show that such a geometric integration can be
constructed substituting appropriately differentials with more general
asymptotic expansions. This can be seen as the basis of geometric integration
similar to that used in geometric measure theory, but without any underlying
differentiable structure, thus allowing Lipschitz functions and rectifiable
sets to be substituted by far less regular objects (e.g. H\"older functions and
their images which may be purely unrectifiable). Our construction includes both
the one-dimensional Young integral and multidimensional integrals introduced
recently by R. Z\"ust. To simplify the exposition, we limit ourselves to
integration of forms of dimensions not exceeding two.
