  A locally conformally Kaehler (l.c.K.) manifold is a complex manifold
admitting a Kaehler covering $\tilde M$, with each deck transformation acting
by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a
holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a
structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold
M is fibered over a circle, the fibers are Sasakian, the fibration is locally
trivial, and M is reconstructed as a Riemannian suspension from the Sasakian
structure on the fibers and the monodromy automorphism induced by this
fibration. This construction is canonical and functorial in both directions.
