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][1])

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?**


  [1]: http://www.ams.org/journals/tran/1987-301-01/S0002-9947-1987-0879582-X/S0002-9947-1987-0879582-X.pdfam=bv.44697112,d.bGE&cad=rja