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Ice sheets and mountain glaciers are critically important components of Earth'sclimate system due to societal and ecological risks associated with sea-level change, ocean freshening, ice-albedo feedback, glacial outburst floods, and freshwater availability. As Earth warms, increasing volumes of surface meltwater will access subglacial environments, potentially lubricating the base of the ice sheets and causing enhanced ice discharge into the ocean. Since subglacial water is effectively hidden beneath the ice, the primary ways to study subglacial hydrological systems are through mathematical modelling and interpreting indirect observations. Glaciers often host subglacial or ice-dammed lakes that respond to changes in subglacial water flow, thereby providing indirect information about the evolution of subglacial hydrological systems. While monitoring subaerial ice-dammed lakes is straightforward, the evolution of subglacial lakes must be inferred from the displacement of the overlying ice surface, posing additional challenges in modelling and interpretation. This dissertation addresses these challenges by developing and analyzing a series of mathematical models that focus on relating subglacial hydrology with observable quantities such as lake level or ice-surface elevation. The dissertation is divided into five chapters. Chapter 1 demonstrates how ageneralization of Nye's (1976) canonical model for subglacial water flow admits a wide class of solitary-wave solutions---localized regions of excess fluid that travel downstream with constant speed and permanent form---when melting at the ice-water interface is negligible. Solitary wave solutions are proven to exist for a wide range of material parameter values that are shown to influence the wave speed and wave profile. Melting at the ice-water interface is shown to cause growth and acceleration of the waves. To relate dynamics like these to observable quantities, Chapter 2 focuses on modelling water-volume oscillations in ice-dammed lakes during outburst flood cycles while accounting for the potential influence of neighboring lakes. Hydraulic connection between neighboring lakes is shown to produce a wide variety of new lake-level oscillations that depend primarily on the relative sizes and proximity of the lakes. In particular, the model produces lake-level time series that mirror ice-elevation changes above a well-known system of Antarctic subglacial lakes beneath the Whillans and Mercer ice streams even though the modelled ice-dammed lakes are not buried beneath the ice. The stability of lake systems with respect to variations in meltwater input is characterized by a transition from oscillatory to steady drainage at high water supply. To create a framework for extending these models of ice-dammed lakes to thesubglacial setting, variational methods for simulating the dynamics of subglacial lakes and subglacial shorelines are derived in Chapter 3. By realizing a direct analogy with the classical Signorini problem from elasticity theory, this chapter also furnishes a new, rigorous computational method for simulating the migration of oceanic subglacial shorelines, which are strongly tied to ice-sheet stability in response to climatic forcings. In Chapter 4, this newly developed model is used to highlight the challenge of accurately interpreting ice-surface elevation changes above subglacial lakes without relying on ice-flow models. The surface expression of subglacial lake activity is shown to depend strongly on the effects of viscous ice flow and basal drag, causing altimetry-derived estimates of subglacial lake size, water-volume change, and apparent highstand or lowstand timing to deviate considerably from their true values under many realistic conditions. To address this challenge, Chapter 5 introduces inverse methods for inferring time-varying subglacial lake activity or basal drag perturbations from altimetry data while accounting for the effects of viscous ice flow. Incorporating horizontal surface velocity data as additional constraints in the inversion is shown to facilitate reconstruction of multiple parameter fields or refinement of altimetry-based estimates. In sum, this dissertation constitutes several novel approaches to understanding ice-water interaction beneath glaciers while laying the foundation for future work seeking to elucidate the role of subglacial processes in the changing climate.