In this thesis we study functions of generators of uniformly bounded semigroups of operators on a Hilbert space. A recent paper of V.V. Peller considers polynomials in an operator T whose iterates (T")n > 0 form a uniformly bounded discrete semigroup. Upper bounds for the norm of a polynomial in T are obtained and both a representation of the Besov space B^-j in B(#) and a von Neumann-type inequality follow. After studying Peller's methods and results, we use a similar approach to study polynomials in two commuting power-bounded operators and obtain comparable norm estimates. These results require a characterisation of Hankel operators on H2(I12), the Hardy space of functions on the two-dimensional torus. We show that the class of such Hankel operators is isometrically isomorphic to the dual of a quotient of a Banach space of operator-valued functions, and we investigate conditions for a generalisation of Nchari's Theorem. Finally, in Chapter 5 we show that analogues of Peller's results hold for functions of the infinitesimal generator of a uniformly bounded, strongly continuous semigroup of operators. This requires a characterisation of the dual space of the injective tensor product L*(R)+)&Ll(R+) using conditional expectation operators, and an identification of the class of Hankel-type integral operator kernels with a subspace of the dual of H'(R).