  A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed
graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k
I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree
$k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair
of vertices connected by an edge in one direction have $\lambda$ common
out-neighbours and a pair of vertices connected by edges in both directions
have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices
can not be connected in both directions.
  We prove that the adjacency matrix of a normally regular digraph is normal. A
connected $k$-regular digraph with normal adjacency matrix is a normally
regular digraph if and only if all eigenvalues other than $k$ are on one circle
in the complex plane. We prove several non-existence results, structural
characterizations, and constructions of normally regular digraphs. In many
cases these graphs are Cayley graphs of abelian groups and the construction is
then based on a generalization of difference sets.
  We also show connections to other combinatorial objects: strongly regular
graphs, symmetric 2-designs and association schemes.
