  We present families of large undirected and directed Cayley graphs whose
construction is related to butterfly networks. One approach yields, for every
large $k$ and for values of $d$ taken from a large interval, the largest known
Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method
yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley
graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially
larger in $k$ than any previously constructed. In the directed case, these are
within a linear factor in $k$ of the Moore bound.
