Wind energy is poised to play a considerable role in the global transition to clean-energy technologies within the next few decades. Modern wind turbines, like aircraft and other aerodynamic structures, are typically designed with the assumption that the flows they encounter will be uniform and steady. However, atmospheric flows are highly unsteady, and systems operating within them must contend with gust disturbances that can lead to performance losses and structural damage. Therefore, the next generation of wind-energy systems requires physics-informed design principles that effectively account for and even leverage these unsteady flow phenomena for enhanced power generation, robustness, and operational longevity. Accordingly, this work details experimental and analytical efforts to characterize unsteady aerodynamics in wind-turbine contexts. In particular, the effects of unsteady streamwise motion on turbine performance are studied, as recent work has suggested that these dynamics may enable time-averaged efficiencies that exceed the steady-flow Betz limit on turbine efficiency. The power production of and flow around a periodically surging wind turbine are thus investigated using wind-tunnel experiments, which suggest that turbines in these flow conditions could leverage unsteady surge motions for power-extraction gains of up to 6.4% over the stationary case. Linearized and nonlinear dynamical models of the response of the turbine to these time-varying flows are derived and validated against the experimental data. These models are also coupled with a potential-flow model of the upstream induction zone of the turbine in order to predict temporal variations in the flow velocities and pressures in this region. Unsteady contributions to the time-averaged efficiency are also considered through theoretical potential-flow derivations. These investigations provide the analytical and experimental foundations for future studies of unsteady atmospheric flows, and will lead to the development of principles and techniques for wind-farm siting, control, and optimization.