  A graph $\G$ admitting a group $H$ of automorphisms acting semi-regularly on
the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over
$H$. Such a graph $\G$ is called {\em normal\/} if $H$ is normal in the full
automorphism group of $\G$, and {\em normal edge-transitive\/} if the
normaliser of $H$ in the full automorphism group of $\G$ is transitive on the
edges of $\G$. % In this paper, we give a characterisation of normal
edge-transitive bi-Cayley graphs, %which form an important subfamily of
bi-Cayley graphs, and in particular, we give a detailed description of
$2$-arc-transitive normal bi-Cayley graphs. Using this, we investigate three
classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups
and metacyclic $p$-groups. We find that under certain conditions, `normal
edge-transitive' is the same as `normal' for graphs in these three classes. As
a by-product, we obtain a complete classification of all connected trivalent
edge-transitive graphs of girth at most $6$, and answer some open questions
from the literature about $2$-arc-transitive, half-arc-transitive and
semisymmetric graphs.
