Provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure, in the infinite dimensional case new phenomena arise. It also provides for the first time a detailed account of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s. Finally, it presents the main ideas and results of the Malliavin calculus.