let $g$ be a finite group and $v$ a finite-dimensional representation of $g$. a {\it compression\/} of $v$ is an equivariant morphism $\phi\colon v \to x$ such that $g$ acts faithfully on the image $\phi(v)$. the main question is the following: \vskip .1in \noindent how far can we compress a given group action, i.e. what is the minimal possible dimension of $x$? \vskip .1in \noindent this minimal dimension depends only on the group $g$ and is called {\it covariant dimension\/} of $g$. for example, if $g$ is commutative, then its covariant dimension equals its rank. but in general, the answer is not known. for the symmetric group $s_n$ there are upper and lower estimates. they were first proved by j. buhler and z. reichstein via the so-called {\it essential dimension\/} of $g$ which is defined similarly to the covariant dimension, but allowing rational compressions $\phi\colon v \to x$. we will introduce the notion of compression and covariant dimension, give a few basic results and discuss somerecent joint work with g.w. schwarz.