The theory of stratified spaces is one of the rapidly developing areas of algebraic topology - an important section of modern mathematical science. This theory already finds numerous applications in such areas of mathematics as the theory of functions of many complex variables, algebraic geometry, variational calculus in general, the theory of differentiated manifolds, etc. The collection "Floored spaces and their applications" is composed of translations of works of foreign scientists (A. Borel, A. Cartana, J. -P. Serra et al . ), introducing the reader into the circle of basic concepts and methods of this branch of science, and is designed to replace the monographs that are not yet available. The book can be useful for both undergraduate and graduate mathematicians, as well as researchers who want to get acquainted with the current state of topology.