  Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of
a metric space (X,d). In the special case that X=G is a finitely generated
group and d is a word metric, the simplest Roe algebra associated to (G,d) is
isomorphic to the reduced crossed product C*-algebra l^\infty(G)\rtimes G.
  Roe algebras are 'coarse invariants', in the sense that if X and Y are
coarsely equivalent metric spaces, then their Roe algebras are isomorphic.
Motivated in part by the coarse Baum-Connes conjecture, we ask if there is a
converse to the above statement: that is, if X and Y are metric spaces with
isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for
very large classes of spaces the answer to this question, and some related
questions, is 'yes'. This can be thought of as a 'C*-rigidity result': it shows
that the Roe algebra construction preserves a large amount of information about
the space, and is thus surprisingly 'rigid'.
  As an example of our results, in the group case we have that if G and H are
finitely generated elementary amenable, hyperbolic, or linear, groups such that
the crossed products l^\infty(G)\rtimes G and l^\infty(H)\rtimes H are
isomorphic, then G and H are quasi-isometric.
