Classification of $k$-forms on $\mathbb{R}^n$ and the existence of associated geometry on manifolds

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on $\mathbb{R}^n$), understood as description of the orbit space of the standard $\mathrm{GL}(n, \mathbb{R})$-action on $\Lambda^k \mathbb{R}^{n*}$ (resp. on $\Lambda ^k \mathbb{R}^n$). 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.