Cyclic homology was introduced in the early eighties independently by
Connes and Tsygan. They came from different directions. Connes wanted to
associate homological invariants to K-homology classes and to describe
the index pair- ing with K-theory in that way, while Tsygan was
motivated by algebraic K-theory and Lie algebra cohomology. At the same
time Karoubi had done work on characteristic classes that led him to
study related structures, without however arriving at cyclic homology
properly speaking. Many of the principal properties of cyclic homology
were already developed in the fundamental article of Connes and in the
long paper by Feigin-Tsygan. In the sequel, cyclic homology was
recognized quickly by many specialists as a new intriguing structure in
homological algebra, with unusual features. In a first phase it was
tried to treat this structure as well as possible within the traditional
framework of homological algebra. The cyclic homology groups were
computed in many examples and new important properties such as prod- uct
structures, excision for H-unital ideals, or connections with cyclic
objects and simplicial topology, were established. An excellent account
of the state of the theory after that phase is given in the book of
Loday.