  Classical Delaunay surfaces are highly symmetric constant mean curvature
(CMC) submanifolds of space forms. We prove the existence of Delaunay-type
hypersurfaces in a large class of compact manifolds, using the geometry of
cohomogeneity one group actions and variational bifurcation techniques. Our
construction specializes to the classical examples in round spheres, and allows
to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging
from complex and quaternionic projective spaces to Kervaire exotic spheres.
