  The theory of lifting classes and the Reidemeister number of single-valued
maps of a finite polyhedron $X$ is extended to $n$-valued maps by replacing
liftings to universal covering spaces by liftings with codomain an orbit
configuration space, a structure recently introduced by Xicot\'encatl. The
liftings of an $n$-valued map $f$ split into self-maps of the universal
covering space of $X$ that we call lift-factors. An equivalence relation is
defined on the lift-factors of $f$ and the number of equivalence classes is the
Reidemeister number of $f$. The fixed point classes of $f$ are the projections
of the fixed point sets of the lift-factors and are the same as those of
Schirmer. An equivalence relation is defined on the fundamental group of $X$
such that the number of equivalence classes equals the Reidemeister number. We
prove that if $X$ is a manifold of dimension at least three, then algebraically
the orbit configuration space approach is the same as one utilizing the
universal covering space. The Jiang subgroup is extended to $n$-valued maps as
a subgroup of the group of covering transformations of the orbit configuration
space and used to find conditions under which the Nielsen number of an
$n$-valued map equals its Reidemeister number. If an $n$-valued map splits into
$n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of
their Reidemeister numbers.
