  We develop a generalized covering space theory for a class of uniform spaces
called coverable spaces. Coverable spaces include all geodesic metric spaces,
connected and locally pathwise connected compact topological spaces, in
particular Peano continua, as well as more pathological spaces like the
topologist's sine curve. Each coverable space has a generalized universal cover
with universal and lifting properties. Associated with this generalized
universal cover is a functorial uniform space invariant called the deck group,
which is related to the classical fundamental group by a natural homomorphism.
We obtain some specific results for one-dimensional spaces. Keywords: universal
cover, uniform space, geodesic space, fundamental group
