This book 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. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject. Table of Contents: Background material; Sobolev spaces on $\mathbb{R}^n$; Differentiable measures on linear spaces; Some classes of differentiable measures; Subspaces of differentiability of measures; Integration by parts and logarithmic derivatives; Logarithmic gradients; Sobolev classes on infinite dimensional spaces; The Malliavin calculus; Infinite dimensional transformations; Measures on manifolds; Applications; References; Subject index. (Surv/164)