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.