  Motivated by the classical bounded H\"{o}lder domains, we introduce the
notion of an unbounded simply connected H\"{o}lder domain. We prove analytic
and geometric characterizations of those domains with the aid of the spherical
metric and the hyperbolic metric. We also study the relation of our definition
to the definition of classical bounded H\"{o}lder domains. It turns out that
unbounded H\"older domains form a natural class of domains for the study of the
Hardy number (which determines the Hardy spaces to which the corresponding
Riemann mapping belongs to). As an application of our characterizations, we
prove a sharp bound for the Hardy number of an unbounded H\"{o}lder domain.
