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.

Introduction.- Basic Concepts of Riemannian Geometry.- Commutations& Variations.- The Weyl Tensor.- Curvature Conditions.- Critical Metrics of Riemannian Functionals.- Bochner-Weitzenböck Formulas and Applications.- Ricci Solitons: Selected Results.- Existence Results of Canonical Metrics on Four Manifolds.- List of Symbols.- References.- Index.