Sobolev spaces and inequalities are fundamental tools in the theory of partial differential equations, analysis, differential geometry, mathematical physics, and more. Introduced 70 years ago, they turned out to be extremely useful in many different settings and continue to attract the attention of new generations of mathematicians. Recent advantages in the theory of Sobolev spaces and in applications are presented by globally recognized specialists in topics covering Sobolev-type spaces of functions in metric spaces, various aspects of Sobolev-type inequalities, boundary value problems for differential operators, spectral problems, approximations, optimal control, important problems of mathematical physics, analysis, partial differential equations, geometry, and more. The book is dedicated to the centenary of S.L. Sobolev and includes biographical articles supplied with the bibliography of Sobolev's works in the 1930s and archive photos of Sobolev previously unpublished in the English-language literature.