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Waves play an essential role in many aspects of plasma science, such asplasma manipulation and diagnostics. Due to the complexity of the governingequations, approximate models are often necessary to describe wave dynamics. Inthis dissertation, waves are treated as geometric objects of a variationaltheory rather than formal solutions of specific PDEs. This approach simplifiescalculations, highlights the underlying wave symmetries, and leads to improvedmodeling of wave dynamics. This thesis presents two breakthroughs that wereobtained in the general theory of waves. The first main contribution is anextension and reformulation of geometrical optics (GO) as a first-principleLagrangian theory that correctly describes polarization effects, such aspolarization precession and the polarization-driven bending of raytrajectories, which appear as leading-order corrections to GO. The theory wasapplied to several systems of interest, such as relativistic spin-1/2 particlesand radio-frequency waves in magnetized plasma. The second main contribution ofthis thesis is the development of a phase-space method to study basicproperties of nonlinear wave--wave interactions. Specifically, I show thatwaves propagating in modulated media, both classical and quantum, canexperience time-averaged refraction caused by effective ponderomotive forces onwave rays. This phenomenon is analogous to the ponderomotive effect encounteredby charged particles in high-frequency electromagnetic fields. I also show thatphase-space methods can be useful to study problems in the field of waveturbulence, such as the nonlinear interaction of high-frequency waves withlarge-scale structures. Overall, the results obtained can serve as a basis forfuture studies on more complex nonlinear wave--wave interactions, such asmodulational instabilities in general wave ensembles or wave turbulence.

A geometric theory of waves and its applications to plasma physics