  We classify, up to orbit equivalence, all cohomogeneity one actions on the
hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the
complex hyperbolic spaces of dimension greater than two. For the quaternionic
hyperbolic spaces of dimension greater than two we reduce the classification
problem to a problem in quaternionic linear algebra and obtain partial results.
For real hyperbolic spaces, this classification problem was essentially solved
by Elie Cartan.
