  Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We
offer a different view of overlays: it resembles the definition of paracompact
spaces via star refinements of open covers. One introduces covering structures
for covering maps and $p:X\to Y$ is an overlay if it has a covering structure
that has a star refinement.
  We prove two characterizations of overlays: one using existence and
uniqueness of lifts of discrete chains, the second as maps inducing simplicial
coverings of nerves of certain covers. We use those characterizations to
improve results of Eda-Matijevi\' c concerning topological group structures on
domains of overlays whose range is a compact topological group.
  In case of surjective maps $p:X\to Y$ between connected metrizable spaces we
characterize overlays as local isometries: $p:X\to Y$ is an overlay if and only
if one can metrize $X$ and $Y$ in such a way that $p|B(x,1):B(x,1)\to
B(p(x),1)$ is an isometry for each $x\in X$.
