  In this paper we survey methods and results of classification of $k$-forms
(resp. $k$-vectors on ${\bf R}^n$), understood as description of the orbit
space of the standard ${\bf GL}(n, {\bf R})$-action on $\Lambda^k {\bf R}^{n*}$
(resp. on $\Lambda ^k {\bf R}^n$). We discuss the existence of related geometry
defined by differential forms on smooth manifolds. This paper also contains an
Appendix by Mikhail Borovoi on Galois cohomology methods for finding real forms
of complex orbits.
