  We consider the degree-diameter problem for undirected and directed circulant
graphs. To date, attempts to generate families of large circulant graphs of
arbitrary degree for a given diameter have concentrated mainly on the diameter
2 case. We present a direct product construction yielding improved bounds for
small diameters and introduce a new general technique for "stitching" together
circulant graphs which enables us to improve the current best known asymptotic
orders for every diameter. As an application, we use our constructions in the
directed case to obtain upper bounds on the minimum size of a subset $A$ of a
cyclic group of order $n$ such that the $k$-fold sumset $kA$ is equal to the
whole group. We also present a revised table of largest known circulant graphs
of small degree and diameter.
