This thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work. In Chapter 2 we study the problem of when P, the set of analytic polynomials, is dense in the Hardy space H^t(G) or the Bergman space L^t_nG, where G is a bounded domain and t ∈ [1,∞). Characterizations of special domains are also given. In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ω. We say a positive measure τ on G is a Carleson measure if there exists a positive constant c such that for each t ∈ [1, ∞) and each polynomial p we have ⎮⎮p⎮⎮_L¹(τ)≤ c ⎮⎮p⎮⎮ _Lᵗ(ω), We characterize all Carleson measures on a normal domain-definition: a domain G where P is dense in H¹(G). It turns out that P is dense in Hᵗ(G) for all t when G is normal. In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation. In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasi similarity under the assumption that G is a normal domain. In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in L²(σ) when R(K) is a hypo dirichlet algebra. In Chapter 7 we illustrate our results by looking at their implications when G' is a crescent. Several interesting function theory problems are studied. In Chapter 8 we study arc length and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let ω be the harmonic measure of G. We show that if J is a rectifiable curve and E ⊂ ∂G ∩ J is a subset with ω(E) > 0, then E has positive length. / Ph. D.