  We study a natural generalization of covering projections defined in terms of
unique lifting properties. A map $p:E\to X$ has the "continuous path-covering
property" if all paths in $X$ lift uniquely and continuously (rel. basepoint)
with respect to the compact-open topology. We show that maps with this property
are closely related to fibrations with totally path-disconnected fibers and to
the natural quotient topology on the homotopy groups. In particular, the class
of maps with the continuous path-covering property lies properly between
Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers.
We extend the usual classification of covering projections to a classification
of maps with the continuous path-covering property in terms of topological
$\pi_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the
continuous path-covering property are classified up to weak equivalence by
subgroups $H\leq \pi_1(X,x_0)$ with totally path-disconnected coset space
$\pi_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation
generated by formally inverting bijective weak homotopy equivalences.
