The last decade has seen parallel developments in computer science and combinatorics, both dealing with networks having strong symmetry properties. Both developments are centred on Cayley graphs: in the design of large interconnection networks, Cayley graphs arise as one of the most frequently used models; on the mathematical side, they play a central role as the prototypes of vertex-transitive graphs. The surveys published here provide an account of these developments, with a strong emphasis on the fruitful interplay of methods from group theory and graph theory that characterises the subject. Topics covered include: combinatorial properties of various hierarchical families of Cayley graphs (fault tolerance, diameter, routing, forwarding indices, etc.); Laplace eigenvalues of graphs and their relations to forwarding problems, isoperimetric properties, partition problems, and random walks on graphs; vertex-transitive graphs of small orders and of orders having few prime factors; distance transitive graphs; isomorphism problems for Cayley graphs of cyclic groups; infinite vertex-transitive graphs (the random graph and generalisations, actions of the automorphisms on ray ends, relations to the growth rate of the graph).

What is the "archetypal" image that comes to mind when one thinks of an infinite graph? What with a finite graph - when it is thought of as opposed to an infinite one? What structural elements are typical for either - by their presence or absence - yet provide a common ground for both? In planning the workshop on "Cycles and Rays" it had been intended from the outset to bring infinite graphs to the fore as much as possible. There never had been a graph theoretical meeting in which infinite graphs were more than "also rans", let alone one in which they were a central theme. In part, this is a matter of fashion, inasmuch as they are perceived as not readily lending themselves to applications, in part it is a matter of psychology stemming from the insecurity that many graph theorists feel in the face of set theory - on which infinite graph theory relies to a considerable extent. The result is that by and large, infinite graph theorists know what is happening in finite graphs but not conversely. Lack of knowledge about infinite graph theory can also be found in authoritative l sources. For example, a recent edition (1987) of a major mathematical encyclopaedia proposes to ". . . restrict [itself] to finite graphs, since only they give a typical theory". If anything, the reverse is true, and needless to say, the graph theoretical world knows better. One may wonder, however, by how much.

The last decade has seen two parallel developments, one in computer science, the other in mathematics, both dealing with the same kind of combinatorial structures: networks with strong symmetry properties or, in graph-theoretical language, vertex-transitive graphs, in particular their prototypical examples, Cayley graphs. In the design of large interconnection networks it was realised that many of the most fre quently used models for such networks are Cayley graphs of various well-known groups. This has spawned a considerable amount of activity in the study of the combinatorial properties of such graphs. A number of symposia and congresses (such as the bi-annual IWIN, starting in 1991) bear witness to the interest of the computer science community in this subject. On the mathematical side, and independently of any interest in applications, progress in group theory has made it possible to make a realistic attempt at a complete description of vertex-transitive graphs. The classification of the finite simple groups has played an important role in this respect."