The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological group isomorphic (in the category of topological groups) to the isometry group of a metric space?
And what about the differentiable version: Is every Liegroup isomorphic (in the category of Liegroups) to a isometry group of a Riemann manifold?
Edit: Benjamin Steinberg gave a reference, which fully answers the question in the topological case. Ryan Budney gave an idea how to realize at least every compact lie group as the isometry group of a Riemannian manifold. (This is proven in THE ISOMETRY GROUPS OF MANIFOLDS AND THE AUTOMORPHISM GROUPS OF DOMAINS, by RITA SAERENS and WILLIAM R. ZAME)
So for me, still open is the question about the non-compact case: Is even every non-compact Lie group realizable as the isometrygroup of a riemannian manifold?