The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments particularly in recent decades it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics astrophysics economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students while Part II contains research papers. Together they represent a valuable resource on both fundamental and advanced aspects of optimal transportation its applications and its interactions with analysis geometry PDE and probability urban planning and economics. Topics covered include Ricci flow the Euler equations functional inequalities curvature-dimension conditions and traffic congestion. Language: English