  Let $G$ be a finite group and $\phi\colon V\to W$ an equivariant morphism of
finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts
faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the
dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. In this paper we
investigate covariant dimension and are able to determine it for abelian groups
and to obtain estimates for the symmetric and alternating groups. We also
classify groups of covariant dimension less than or equal to 2. A byproduct of
our investigations is the existence of a purely transcendental field of
definition of degree $n-3$ for a generic field extension of degree $n\geq 5$.
