This book focuses on a selection of special topics, with emphasis on past and present research of the authors on ?canonical ? Riemannian metrics on smooth manifolds. On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of ?Curvature Conditions ? and ?Critical Metrics ? of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.