  In this paper we study compact Sasaki manifolds in view of transverse
K\"ahler geometry and extend some results in K\"ahler geometry to Sasaki
manifolds. In particular we define integral invariants which obstruct the
existence of transverse K\"ahler metric with harmonic Chern forms. The integral
invariant $f_1$ for the first Chern class case becomes an obstruction to the
existence of transverse K\"ahler metric of constant scalar curvature. We prove
the existence of transverse K\"ahler-Ricci solitons (or {\it 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
irregular toric Sasaki-Einstein metrics on the unit circle bundles of the
powers of the canonical bundle of the two-point blow-up of the complex
projective plane.
