In this talk, we discuss a program concerning the construction, computation, and properties of solutions of evolution and wave equations using techniques from harmonic analysis, in particular, localization in phase space, following the dyadic parabolic decomposition leading to a multi-scale approach. We begin with the representation of Fourier integral operators associated with canonical graphs and develop explicit separated expansions of the complex exponential in the oscillatory integral representations of their kernels; we obtain a low separation rank using the dyadic parabolic decomposition of phase space and prolate spheroidal wave functions. This expansion leads to the development of a fast algorithm. We show an application in seismic inverse scattering. Our algorithm also provides the basis of a computational procedure following the construction of weak solutions of Cauchy initial value problems for the wave equation if the wavespeed is C^{2,1}, in which, in addition, a Volterra equation needs to be solved. The general construction in a medium with Holder regularity s>=2 is initiated by the construction of an approximate solution following the smoothing, that is, paradifferential decomposition of the symbol of the evolution or wave operator, and is completed by solving a Volterra equation of the second kind which corrects for the symbol smoothing and essentially accounts for the scattering between wave packets. We establish regularity estimates in the evolution coordinate for the Volterra kernel and solution of the Volterra equation. These estimates govern the choice of quadrature used when solving the Volterra equation, and subsequently the initial value problem, numerically. Our approach and analysis aids in the understanding of the notion of scale in the wave field and how this interacts with a medium, and leads to precise estimates expressing the degree of concentration of wave packets. Furthermore, we establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lame' parameters. Joint research with S. Holman, H. Smith, G. Uhlmann, A. Vasy and H. Wendt.