  In this paper, we introduce a method for finding all edge-transitive graphs
of small order, using faithful representations of transitive permutation groups
of small degree, and we explain how we used this method to find all
edge-transitive graphs of order up to $47$, and all bipartite edge-transitive
graphs of order up to $63$. We also give an answer to a 1967 question of
Folkman about semi-symmetric graphs of large valency; in fact we show that for
semi-symmetric graphs of order $2n$ and valency $d$, the ratio $d/n$ can be
arbitrarily close to $1$.
