  We explore the geometry of nonpositively curved spaces with isolated flats,
and its consequences for groups that act properly discontinuously, cocompactly,
and isometrically on such spaces. We prove that the geometric boundary of the
space is an invariant of the group up to equivariant homeomorphism. We also
prove that any such group is relatively hyperbolic, biautomatic, and satisfies
the Tits Alternative. The main step in establishing these results is a
characterization of spaces with isolated flats as relatively hyperbolic with
respect to flats. Finally we show that a CAT(0) space has isolated flats if and
only if its Tits boundary is a disjoint union of isolated points and standard
Euclidean spheres.
  In an appendix written jointly with Hindawi, we extend many of the results of
this article to a more general setting in which the isolated subspaces are not
required to be flats.
