  We study renormalization group flow in a non-local version of quantum
electrodynamics (QED). We determine the regime in which the theory flows to a
local theory in the infrared and study a possible UV completion of
four-dimensional QED. In addition, we find that there exist non-local conformal
theories with a one-dimensional conformal manifold and non-local deformations
of QED in three dimensions that are exactly marginal. Along the way we develop
methods for coupling non-local derivatives to external sources and discuss
unitarity and conformal vs. scale invariance of these theories.
