This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independtly from the work of Alain Connes and the author. They were motivated by considerations of operator algebras and algebraic K-theory. The good framework to develop these considerations is cyclic homology (or non commutative De Rham homology). This enables us to extend the classical Chern-Weil theory far beyond his original scope (at least for the general linear group). Cyclic homology is the natural target for characteristic classes and its computa-tion is a matter of homological algebra as was shown by A. Connes. On the other hand, the objets we are taking the characteristic classes of are elements of the K-theory of a ring. This K-theory (algebraic or topological) is difficult to compu-te in general. Cyclic homology (also called "additive K-theory" by Feigin and. Tsygan) appears therefore as a first step to compute the K-groups for general al-gebras. We have tried to make this book as self-contained as possible. Together with the motivations provided by A. Connes, the reader should not find any special difficul-ty to read it. In particular, the first chapters can be easily integrated in a gra-duate course on the subject.