  Ramanujan graphs have fascinating properties and history. In this paper we
explore a parallel notion of Ramanujan digraphs, collecting relevant results
from old and recent papers, and proving some new ones. Almost-normal Ramanujan
digraphs are shown to be of special interest, as they are extreme in the sense
of an Alon-Boppana theorem, and they have remarkable combinatorial features,
such as small diameter, Chernoff bound for sampling, optimal covering time and
sharp cutoff. Other topics explored are the connection to Cayley graphs and
digraphs, the spectral radius of universal covers, Alon's conjecture for random
digraphs, and explicit constructions of almost-normal Ramanujan digraphs.
