In this thesis we shall be considering a variety of problems set in the complex plane whose common feature is that they involve domains of finite multiple connectivity. We choose to focus on a particular canonical class of domains, namely circular domains. We extend our results to more general domains using conformal mappings. Results are derived for these circular domains by using the theory of Schottky groups. Problems we consider include the construction of automorphic functions, Green’s functions, and conformal mappings from circular domains to other commonly studied canonical domains. The abstract function-theoretic results we derive are applied to a number of physical problems of fluid dynamics