When Pontryagin established the duality between discrete and compact abelian groups in 1932 he was motivated by rather specific applications, mostly arising in an attempt at a general theory relating the following two examples from algebraic topology. Cech^s homology groups of a compact space appeared as inverse limits of homology groups of finite complexes and thus behaved like compact abelian groups, whereas the discrete Cech cohomology groups arose from direct limits. The duality theory, however, evolved rather quickly to a rich structure theory which was applied to numerous areas of algebra, topology and analysis. In algebra and number theory these applications reach from Pontryagin*s classification of the locally compact connected fields to the modern presentation of algebraic number theory (see W-1) while in group theory itself a rich interplay between the theory of abelian groups and compact groups developed giving impulses to both lines of research. Harmonic analysis, which had seen a great deal of activity during the twenties, was provided with precisely the right abstract tools by Pontryagin duality theory, and harmonic analysis became inseparable from the duality of locally compact abelian groups. Other dualities for various classes of topological groups followed, exemplified by the work of Tannaka and Krein in the thirties and forties, and the process of finding duality theories for general locally compact groups is still not completed.