In this thesis we focus on the relation between random matrix theory and orthogonal polynomial theory in the complex plane. It is well known that even if the entries of a random matrix are independent, the eigenvalues will be highly correlated. This corre- lation, which is a pairwise logarithmic repulsion between the eigenvalues, leads one to think that the eigenvalues of a random matrix behave like particles in a Coulomb gas, since the logarithmic repulsion is the Coulomb interaction in two dimensions.<br />
We consider the case when the particles are confined to an ellipse in the plane. At inverse temperature beta equal 2, we introduce new families of exactly solvable two-dimensional Coulomb gases for a fixed and finite number of particles N. We find, in the analysis of local fluctuations in the weak non-Hermiticity limit of the correlation functions, old and new universality classes. This is achieved by showing that certain subfamilies of Jacobi polynomials extend to orthogonality relations over a weighted ellipse in the plane.