  Classification results are given for (i) compact quaternionic K\"ahler
manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain
complete hyperK\"ahler manifolds with a cohomogeneity-two action of a
semi-simple group preserving each complex structure, (iii) compact 3-Sasakian
manifolds which are cohomogeneity one with respect to a group of 3-Sasakian
symmetries. Information is also obtained about non-compact quaternionic
K\"ahler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits
in complex semi-simple Lie algebras.
