Einstein metrics and GIT stability

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