<p>Motivated by the classical bounded Hölder domains, we introduce the notion of an unbounded simply connected Hölder domain. We prove analytic and geometric characterizations of these 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 Hölder domains. It turns out that unbounded Hölder 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 Hölder domain.</p>