This book demonstrates the development of the regularity theory of solutions of fully nonlinear elliptic equations. Caffarelli and Cabre offer a detailed presentation of all techniques needed to extend the classic Schauder and Calderon-Zygmund regularity theory for linear elliptic equations to a fully nonlinear context. The authors present key ideas and prove all results needed for the theory of viscosity solutions of nonlinear equations, and regularity theory for convex fully nonlinear equations and for fully nonlinear equations with variable coefficients. This book is suitable as a text for graduate courses in nonlinear elliptic partial differential equations.