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This textbook is addressed to PhD or senior undergraduate students
in mathematics, with interests in analysis, calculus of variations,
probability and optimal transport. It originated from the teaching
experience of the first author in the Scuola Normale Superiore,
where a course on optimal transport and its applications has been
given many times during the last 20 years. The topics and the tools
were chosen at a sufficiently general and advanced level so that
the student or scholar interested in a more specific theme would
gain from the book the necessary background to explore it. After a
large and detailed introduction to classical theory, more specific
attention is devoted to applications to geometric and functional
inequalities and to partial differential equations.
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