  We study the Lie algebra of infinitesimal isometries on compact Sasakian and
K--contact manifolds. On a Sasakian manifold which is not a space form or
3--Sasakian, every Killing vector field is an infinitesimal automorphism of the
Sasakian structure. For a manifold with K--contact structure, we prove that
there exists a Killing vector field of constant length which is not an
infinitesimal automorphism of the structure if and only if the manifold is
obtained from the Konishi bundle of a compact pseudo--Riemannian
quaternion--Kaehler manifold after changing the sign of the metric on a maximal
negative distribution. We also prove that non--regular Sasakian manifolds are
not homogeneous and construct examples with cohomogeneity one. Using these
results we obtain in the last section the classification of all homogeneous
Sasakian manifolds.
