Einstein metrics and GIT stability

In this expository article we review the problem of finding Einstein metrics on compact Kähler manifolds and Sasaki manifolds. In the former half of this article we see that, in the Kähler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal Kähler metrics or constant scalar curvature Kähler (cscK) metrics. In the latter half of this paper we see that most of ideas in Kähler geometry extend to Sasaki geometry as transverse Kähler geometry. We also summarize recent results about the existence of toric Sasaki-Einstein metrics.