Explore the fundamental concepts of optimal transportation, Monge-Ampère equations, and Riemannian Ricci bounds in this comprehensive lecture delivered by L. Ambrosio from Scuola Normale Superiore di Pisa, Italy. Delve into the intricate connections between these mathematical topics as the speaker presents the first part of a series on this subject. Gain valuable insights into the theoretical foundations and practical applications of these advanced mathematical concepts in the fields of geometry, analysis, and probability theory. This in-depth presentation, lasting nearly two hours, offers a rigorous exploration of the subject matter, suitable for graduate students and researchers in mathematics and related disciplines.