  In this paper, using connections between Clifford-Wolf isometries and Killing
vector fields of constant length on a given Riemannian manifold, we classify
simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get
the classification of complete simply connected Riemannian manifolds with the
Killing property defined and studied previously by J.E. D'Atri and H.K.
Nickerson. In the last part of the paper we study properties of
Clifford-Killing spaces, that is, real vector spaces of Killing vector fields
of constant length, on odd-dimensional round spheres, and discuss numerous
connections between these spaces and various classical objects.
