Matrix theories of elastic and acoustic wave scattering are reviewed and unified, and a new one is devised and discussed. Called MOOT (method of optimal truncation), it has tangible and aesthetic advantages over other methods, particularly its convergence properties and its conceptual straightforwardness. The exposition is, for simplicity, in terms of scalar waves; the following paper contains detailed applications to scattering of elastic waves. A family of matrix equations, which includes the present method and others, is derived in a simple way from the boundary conditions. Integral equations and their solution by matrix methods are discussed, MOOT is developed and compared with other matrix methods, symmetry principles are developed and their enforcement discussed, and certain computational methods, details, and limitations are expounded. Briefly, we proceed by expanding the scattered wave in a truncated series of eigenfunctions of the unperturbed wave equation, and determine the expansion coefficients (scattered amplitudes) by requiring that the mean square of the deviance (discontinuity in value or normal derivative in the scalar case) from the boundary conditions at the surface of the scatterer be minimized. This results in matrix equations for the scattered amplitudes which may, in many cases, be easily solved. The method is useful for computing scattering of acoustic, elastic, or electromagnetic waves from defects which are internally piecewise homogeneous, so that conditions on the wave function derivatives and values at the boundaries characterize the scatterers. Although the method is applicable to general shapes, the computations are accelerated if the scatterers are axially symmetric. The matrix equations are superficially similar to others derived and used in the past, which were not based on an optimization principle. Differences are exhibited and their significance is discussed.