More than one definition has been used for the word “normal” as applied to Cayley graphs, but several authors have recently used the following definition: a Cayley graph for a group

$G$ is said to be “normal” if

$G$ (in its regular action) is normalized by the full automorphism group of the graph. In the present paper the author defines a Cayley graph to be “normal edge-transitive” if its automorphism group contains a subgroup which both normalizes

$G$ and acts transitively on the edges. (Throughout the paper the author considers both “graphs,” by which she means directed graphs, and “undirected graphs.”) The author suggests that these form a subfamily of central importance to analyzing the family of all Cayley graphs for a given group. For example, she proves that any Cayley graph for

$G$ is an edge-disjoint union of normal edge-transitive Cayley graphs for

$G$. The author also proves results on quotients of Cayley graphs (describing when these are Cayley graphs for a quotient group, etc.), and on constructing all normal edge-transitive Cayley graphs having a given normal edge-transitive Cayley graph as a quotient. The paper concludes with a sample theorem about the full automorphism groups of normal edge-transitive Cayley graphs for finite simple groups.