Abstract
This paper gives a survey of results on Hermitian K-theory over the last ten years. The main emphasis is on the computation of the numerical invariants of Hermitian forms with the help of the representation theory of discrete groups and by signature formulae on smooth multiply-connected manifolds. In the first chapter we introduce the basic concepts of Hermitian K-theory. In particular, we discuss the periodicity property, the Bass-Novikov projections and new aids to the study of K-theory by means of representation spaces. In the second chapter we discuss the representation theory method for finding invariants of Hermitian forms. In § 5 we examine a new class of infinite-dimensional Fredholm representations of discrete groups. The third chapter is concerned with signature formulae on smooth manifolds and with various problems of differential topology in which the signature formulae find application.