Algebraic varieties are shapes defined by polynomial equations.Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres.They belong to 105 irreducible deformation families.This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago.The book's solution exploits the relation between these metrics and the algebraic notion of K-stability.Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces.This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.