Abstract. Let G be an infinite finitely generated group. We focus on a new invariant R(G) with values in [0, 1], called Hilbert space compression. It describes how close any uniform embedding of the group into a Hilbert space can be to a quasi-isometry. Using infinite families of expanders, we construct (explicitly) groups which are uniformly embeddable into Hilbert space but, like the famous Gromov random groups, have zero Hilbert space compression. We extend our results to embeddings into Banach spaces.