  This is a survey paper, starting from the general notion of coordinate bundle
taken from Steenrod. Its aim is to provide a motivation for the introduction of
cyclic homology (and the closely related noncommutative de Rham cohomology) by
Connes, Tsygan and the author. The bridge is made through a generalization of
Chern-Weil theory, explained in a very concrete manner through the transition
functions of the bundles involved. The final result is a "Chern character"
introduced by Connes and the author in the framework of K-theory and functional
analysis.
