Algebraic $K$-theory of rings from a topological viewpoint

Because of its strong interaction with almost every part of pure mathematics, algebraic K-theory has had a spectacular development since its origin in the late fifties. The objective of this paper is to provide the basic definitions of the algebraic K-theory of rings and an overview of the main classical theorems. Since the algebraic K-groups of a ring R are the homotopy groups of a topological space associated with the general linear group over R, it is obvious that many general results follow from arguments from homotopy theory. This paper is essentially devoted to some of them: it explains in particular how methods from stable homotopy theory, group cohomology and Postnikov theory can be used in algebraic K-theory.