The work presented in this thesis is particularly concerned with a robust integral equation formulation of acoustic scattering and radiation problems, which are essentially exterior Neumann boundary-value problems. Both layer theory and the Helmholtz formula, used in the classical formulation (pre- 1968), result in a non-uniqueness problem. This non-uniqueness is purely mathematical and has no bearing on the actual physical problem. Various workers over the past two decades or so developed alternative formulations, which resolve the problem of non-uniqueness but also suffer from computational drawbacks. Kussmaul (1969) developed a formulation involving the superposition of a simple-layer potential and a double-layer potential, combined by a coupling parameter. Kussmaul also presented a uniqueness proof valid for all wave- numbers. However his formulation involves an integral operator which has a hypersingular kernel. This creates computational difficulties. My thesis presents a new integral equation formulation which involves the superposition of a layer potential generated by simple sources on the given boundary, plus a layer potential generated by dipole sources located on an interior boundary similar and similarly situated to the given boundary. These two potentials are also combined by a coupling parameter. However, unlike the Kussmaul formulation, this avoids the integral operator containing the hypersingular kernel. An argument towards uniqueness is presented. Some test radiation problems and some scattering problems are investigated. Numerical results are given which show that the new formulation gives excellent agreement with the analytical results. The thesis also presents a derivation of wave-functions via layer potentials generated by a uniform distribution of sources on a spherical surface. This is utilized in the discussion of the hypersingular kernel of a certain integral operator, and the analysis is used to verify Terai's (1980) result for a hypersingular integral on a flat plate.