  We classify curvature homogeneous hypersurfaces in S^4 and H^4. In higher
dimesnsion one only has the FKM examples and an isolate one by Tsukada of a
hypersurface in H^5.
  Besides some simple examples, we show that there exists an isolated
hypersurface with a circle of symmetries and and a one parameter family
admitting no continuous symmetries. Outside the set of minimal points, which
only exists in the case of S^4, every example is locally and up to covers of
this form.
