Let [omega] be a bounded, simply connected domain in C with [partial][omega] analytic. Assume that 0ϵ[omega]. Let S([omega]) denote the class of functions F(z) which are analytic and univalent in [omega] with F(0) = 0 and F[superscript]'(0) = 1. Let [phi][subscript]n(z)\[subscript]spn=0[infinity] be the Faber polynomials associated with [omega]. If F(z)ϵ S([omega]) then F(z) can be expanded in a Faber series of the form F(z) = [sigma][limits][subscript]spn=0[infinity]A[subscript]n[phi][subscript]n(z), zϵ[omega].LetE = \x+iy:x[superscript]2[over](5/4)[superscript]2 + y[superscript]2[over](3/4)[superscript]2 < 1.;In Chapter 2, we obtain sharp bounds for the Faber coefficients A[subscript]0, A[subscript]1 and A[subscript]2 of functions F(z) in S(E) and in certain related classes. In addition. we find sharp bounds for certain linear combinations of the Faber coefficients for the same class of functions;In Chapter 3, we obtain global sharp bounds for the Faber coefficients of functions F(z) in certain related classes and subclasses of S(E).