The work consists of three chapters. After the brief Chapter 1 ("General Theory"), we begin the systematic computation of the cohonology of infinite-dimensional Lie algebras in Chapter 2. The main results of this chapter concern the algebras of formal and smooth vector fields, current algebras, and Kac-Moody algebras. (The first and last sections of this chapter deal with another topic: the cohomology of finite-dimensional Lie algebras and the cohonology of Lie superalgebras; the latter, in their methods, results, and applications, are fairly close to the homology of infinite-dimensional Lie algebras.) The concluding chapter is devoted to applications. These applications comprise the characteristic classes of foliations, combinatorial identities known as the Hacdonald identities, invariant differential operators, cohomology, and, in particular, central extensions of Lie groups and cohomology operations in cobordism theory.