  These lecture notes contain an exposition of basic ideas of K-theory and
cyclic cohomology. I begin with a list of examples of various situations in
which the K-functor of Grothendieck appears naturally, including the rudiments
of the topological and algebraic K-theory, K-theory of C^*-algebras, and
K-homology. I then discuss elementary properties of cyclic cohomology using the
Cuntz-Quillen version of the calculus of noncommutative differential forms on
an algebra. As an example of the relation between the two theories we describe
the Chern homomorphism and various index-theorem type statements. The remainder
of the notes contains some more detailed calculations in cyclic and reduced
cyclic cohomology. A key tool in this part is Goodwillie's theorem on the
cyclic complex of a semi-direct product algebra. The final chapter gives an
exposition of the entire cyclic cohomology of Banach algebras from the point of
view of supertraces on the Cuntz algebra. The results discussed here include
the simplicial normalization of the entire cyclic cohomology, homotopy
invariance and the action of derivations.
