In this paper we study compact Sasaki manifolds in view of
transverse Kähler geometry and extend some results in Kähler
geometry to Sasaki manifolds. In particular we define integral invariants
which obstruct the existence of transverse Kähler metric
with harmonic Chern forms. The integral invariant $f1$ for the first
Chern class case becomes an obstruction to the existence of transverse
Kähler metric of constant scalar curvature. We prove the
existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton)
on compact toric Sasaki manifolds whose basic first Chern
form of the normal bundle of the Reeb foliation is positive and
the first Chern class of the contact bundle is trivial. We will
further show that if $S$ is a compact toric Sasaki manifold with
the above assumption then by deforming the Reeb field we get
a Sasaki-Einstein structure on S. As an application we obtain
Sasaki-Einstein metrics on the $U(1)$-bundles associated with the
canonical line bundles of toric Fano manifolds, including as a special
case an irregular toric Sasaki-Einstein metrics on the unit circle
bundle associated with the canonical bundle of the two-point
blow-up of the complex projective plane.