This lecture will survey progress over the past several years in the study of Einstein metrics, and metrics having two and one sided bounds on Ricci curvature, on compact manifolds. Discussion will begin with results on the moduli space of Einstein metrics, and its compactifications, in particular on 4-manifolds. Recent progress on convergence and degeneration of metrics with bounds on Ricci curvature by Perelman, Colding, Cheeger and the speaker will also be presented.
A number of open problems will be discussed, reflecting the current state in the field. The relevance of this recent progress toward the basic existence of Einstein metrics, and other metrics satisfying curvature equations, will be indicated.