The scattering of acoustic waves by boundary discontinuities in waveguides is analyzed using the Method of Matched Asymptotic Expansions (MAE). Existing theories are accurate only for very low frequencies. In contrast, the theory developed in this thesis is valid over the entire range of frequencies up to the first cutoff frequency. The key to this improvement lies in recognizing the important physical role of the cutoff cross-modes of the waveguide, which are usually overlooked. Although these modes are evanescent, they contain information about the interaction between the local field near the discontinuity and the far-field. This interaction has a profound effect on the far-field amplitudes and becomes increasingly important with frequency. The cutoff modes also present novel mathematical problems in that current asymptotic techniques do not offer a rational means of incorporating them into a mathematical description. This difficulty arises from the non-Poincare form of the cross-modes, and its resolution constitutes the second new result of this thesis. We develop a matching scheme based on block matching intermediate expansions in a transform domain. The new technique permits the matching of expansions of a more general nature than previously possible, and may well have useful applications in other physical situations where evanescent terms are important. We show that the resulting theory leads to significant improvements with just a few cross-mode terms included, and also that there is an intimate connection with classical integral methods. Finally, the theory is extended to waveguides with slowly varying shape. We show that the usual regular perturbation analysis of the wave regions must be completely abandoned. This is due to the evanescent nature of the cross-modes, which must be described by a WKB approximation. The pressure field we so obtain includes older results. The new terms account for the cutoff cross-modes of the variable waveguide, which play a central role in extending the dynamic range of the theory.