Homogeneous spaces of infinite-dimensional Lie algebras and characharacteristic classes of foliations

In this article we introduce a new language to describe many problems of differential geometry: for example, problems connected with the theory of pseudogroups, Lie equations, foliations, characteristic classes, etc. This is the language of infinite-dimensional Lie algebras and their homogeneous spaces. It is closely connected with the general idea of formal differential geometry set forth by I. M. Gel'fand in his lecture at the International Congress of Mathematicians in Nice. In addition to a detailed account of the theory of homogeneous spaces of infinite-dimensional Lie algebras, this article contains applications of this theory to the characteristic classes of foliations. It also includes results on these questions from earlier papers by I. M. Gel'fand, D. A. Kazhdan, D. B. Fuks, and the authors.