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Ray optics is an intuitive and computationally efficient model for wave propagation through nonuniform media. However, the underlying geometrical-optics (GO) approximation of ray optics breaks down at caustics, erroneously predicting the wave intensity to be infinite and thereby limiting the predictive capabilities of GO-based codes. Full-wave modeling can be used instead, but the added computational cost brings its own set of tradeoffs. Developing more efficient caustic remedies has therefore been an active area of research for the past few decades. In this thesis, I present a new ray-based approach called 'metaplectic geometrical optics' (MGO) that can be applied to any linear wave equation. Instead of evolving waves in the usual x (coordinate) or k (spectral) representation, MGO uses a mixed X=Ax+B representation. By continuously adjusting the coefficients A and B along the rays via sequenced metaplectic transforms (MTs) of the wavefield, corresponding to symplectic transformations of the ray phase space, one can ensure that GO remains valid in the X coordinates without caustic singularities. The caustic-free result is then mapped back onto the original x space using metaplectic transforms, as demonstrated on a number of examples. Besides outlining the basic theory, this thesis also presents specialized fast algorithms for MGO. These algorithms focus on the MT, which is a unitary integral mapping that can be considered a generalization of the Fourier transform. First, a discrete representation of the MT is developed that can be computed in linear time when evaluated in the near-identity limit; finite MTs can then be implemented by iterating $K\gg 1$ near-identity MTs. Second, an algorithm based on Gauss--Freud quadrature is developed for efficiently computing finite MTs along their steepest-descent curves, which may be useful in catastrophe-optics applications beyond MGO.