The study of wave processes in fluid-saturated porous media represents a fundamental problem in continuum mechanics, holding key significance for geophysics, seismic exploration, hydrogeology, and acoustics. Over the past seven decades, significant research efforts have been directed toward developing mathematical models describing coupled deformation and filtration processes in such media. This review systematizes and analyzes the main stages in the development of the theory of wave processes in deformable fluid-saturated porous media, starting from the foundational works of Maurice Biot and extending to modern generalizations incorporating anisotropy, viscoelasticity, relaxation effects, and stochastic parameter variability. Special attention is given to the comparative analysis of constitutive relations, wave types (fast and slow compressional, shear), interface boundary conditions, and numerical methods for implementing the considered models. Current trends in the development of the theory are identified, including the application of fractional calculus to account for memory effects in porous media and stochastic approaches for quantifying parameter uncertainties.