This thesis constitutes a study in the field of approximation theory and is restricted to sets defined on the complex plane. The main objective is to present means that have been used to determine whether a sequence of polynomials can be considered as being uniformly convergent to a given analytic function and if convergence can be considered stronger than uniform. Background material is given in Chapter I. This includes definitions of point sets and of measures of approximation. Also basic theorems concerning both approximation and conformal mapping are given. In Chapter II properties of conformal mappings are established. The theorems discussed lead to a statement of necessary and sufficient conditions for a mapping of a simply connected region onto the interior of the unit circle to be homeomorphic on the closure of the region. The bulk of the work presented in Chapter II is based on definitions and theorems given by Caratheodory and Markushevich. The last chapter puts to use the theorems given in Chapters I and II to prove Walsh's Theorem and Farrell's Theorem. Other theorems originally presented by Walsh and Mergelyan are also discussed in Chapter III. The thesis concludes with examples of sequences uniformly convergent but not convergent in a given measure of approximations in order to show the reader that the latter property is stronger. / M.S.