In Chapter 2, we derive the formulations of scattering, absorption and extinction coefficients, and the phase matrix in the VRT equation for discrete scatterers ( spherical Rayleigh and Mie particles, non-spherical particles such as small spheroids, disks and needles, and cylinders) and continuous random media. The correlation function and correlation length of random media are also discussed. The boundary conditions of VRT approach for stratified medium are derived.In Chapter 3, we list some physical parameters in the microwave region for atmosphere, soil, water and sea water, ice and sea ice, vegetation materials etc, which are usually used in the numerical modelling, such as the dielectric constant and the size distribution.In Chapter 4, we discuss several approaches to the VRT equation, e. Preface ix and application in SAR imaging are also discussed.In Chapter 5, we discuss the models of inhomogeneous random media and the approaches for coupled VRT equations of multi-layer random media, a layer of random clusters of non-spherical scatterers, and two-dimensional VRT equation of non-uniform scatterer medium. The applications in polarimetric scattering and thermal emission of crops and forestry, and atmospheric precipitation are discussed.To develop the basic theory of electromagnetic scattering beyond Maxwell's equations, we need to look at the analytic wave theory of random media, which describes the basic principles of electromagnetic scattering and the VRT approach. In Chapter 6, using the mean dyadic Green's function (DGF) of stratified media, we derive the first Another important topic of random media is the scattering theory from randomly rough surfaces. In Chapter 9, we introduce main conclusions using the Kirchhoff approximation (KA) and the geometric solution, and the small perturbation theory (SPA), and derive the shadowing function for high-order scattering.