  We define a covering of a profinite graph to be a projective limit of a
system of covering maps of finite graphs. With this notion of covering, we
develop a covering theory for profinite graphs which is in many ways analogous
to the classical theory of coverings of abstract graphs. For example, it makes
sense to talk about the universal cover of a profinite graph and we show that
it always exists and is unique. We define the profinite fundamental group of a
profinite graph and show that a connected cover of a connected profinite graph
is the universal cover if and only if its profinite fundamental group is
trivial.
