  Vaisman manifolds are strongly related to K\"ahler and Sasaki geometry. In
this paper we introduce toric Vaisman structures and show that this
relationship still holds in the toric context. It is known that the so-called
minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki
manifold. We show that if a complete Vaisman manifold is toric, then the
associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki
manifold, whose K\"ahler cone is equipped with an appropriate compatible
action, gives rise to a toric Vaisman manifold. In the special case of a
strongly regular compact Vaisman manifold, we show that it is toric if and only
if the corresponding K\"ahler quotient is toric.
