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Plasma wakefields, whether driven by laser or particle bunch, possess a stable, coherent, and robust structure that can accelerate particles to high energy. Wakefields derive these remarkably properties from a single physical principle: waves with a phase velocity much higher than the bulk speed of a population particles will not strongly couple to the particles. Instead, a small subset of particles are captured by the wake and coherently accelerated to high energy. In typical applications of wakefield acceleration, this property is harnessed to accelerate electrons, but extends widely into other applications. Notably, ion-cyclotron waves in fusion plasmas, when possessed of a fast phase velocity, can generate a fast-ion tail that can dramatically enhance fusion yield without increasing the bulk ion temperature, as has been observed in the C-2U experiment. The first part of this work explores this phenomenon, examining the physics of beam-driven ion-cyclotron waves in the context of the scrape-off-layer (SOL) of a field-reversed-configuration (FRC) plasma. In particular, the nonlinear mechanics of ion acceleration by these waves is examined. Interestingly, this principle of phase velocity can be just as useful when violated. In the field of laser-wakefield acceleration (LWFA), in the regime of near-critical density plasmas, the physics of wakefield acceleration transitions into a sheath acceleration, which is potentially capable of generating a large flux of low-energy electrons. The application of this principle to a novel method of internal cancer therapy is treated in the second part of this work.