Abstract: In a recent work with Florit and Ziller, we completed the classification of curvature-homogeneous hypersurfaces in spaces of constant curvature, treating the one remaining unsolved case, that of a hypersurface in a 4-dimensional space form. It was a surprise to discover that, in this case, there exists an ‘exotic’ family of solutions that are not homogeneous as hypersurfaces, and it turns out that a variety of techniques are needed to analyze them fully.