  The generalised Paley graphs are, as their name suggests, a generalisation of
the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise
as the relation graphs of symmetric cyclotomic association schemes. However,
their automorphism groups may be much larger than the groups of the
corresponding schemes. We determine the parameters for which the graphs are
connected, or equivalently, the schemes are primitive. Also we prove that
generalised Paley graphs are sometimes isomorphic to Hamming graphs and
consequently have large automorphism groups, and we determine precisely the
parameters for this to occur. We prove that in the connected, non-Hamming case,
the automorphism group of a generalised Paley graph is a primitive group of
affine type, and we find sufficient conditions under which the group is equal
to the one-dimensional affine group of the associated cyclotomic association
scheme. The results have been applied in \cite{LLP} to distinguish between
cyclotomic schemes and similar twisted versions of these schemes, in the
context of homogeneous factorisations of complete graphs.
