  We present the current results in the study of weighted composition operators
on weighted Banach spaces of an unbounded, locally finite metric space.
Specifically, we determine characterizations of bounded and compact weighted
composition operators, including the operator and essential norms. In addition,
we characterize the weighted composition operators that are injective, are
bounded below, have closed range, and have bounded inverse. We characterize the
isometries and surjective isometries among the weighted composition operators,
as well as those that satisfy the Fredholm condition. Lastly, we provide
numerous interesting examples of the richness of these operators acting on the
discrete weighted Banach spaces.
