In this paper, part HI of a two paper set, we place Green’s theorem based reverse time migration (RTM), for the first time on a firm footing and technically consistent math-physics foundation. The required new Green’s function for RTM application is developed and provided, and is neither causal, anticausal, nor a linear combination of these prototype Green’s functions, nor these functions with imposed boundary conditions. We describe resulting fundamentally new RTM theory and algorithms, and provide a step-by-step prescription for application in 1D, 2D and 3D, the fatter for an arbitrary laterally and vertically varying velocity field. The original RTM method of running the wave equation backwards with surface reflection data as a boundary condition is not a wave theory method for wave-field prediction, neither in depth nor in reversed time. In fact, the latter idea corresponds to Huygens Principle which evolved and was corrected and became a wave theory predictor by George Green in 1826. The original RTM method, where (1) ‘running the wave equation backward in time’, and then (2) employing a zero lag cross-correlation imaging condition, is in both of these ingredients less accurate and effective than the Green’s theorem RTM method of this two paper set. Furthermore, all currently available Green’s theorem methods for RTM make fundamental conceptual and algorithmic errors in their Green’s theorem formulations. Consequently, even with an accurate velocity model, current Green’s theorem RTM formulations can lead to image location errors and other reported artifacts. Addressing the latter problems is a principal goal of the new Green’s theorem RTM method of this paper. Several simple analytic 1D examples illustrate the new RTM method, We also compare the general RTM methodology and philosophy, as the high water mark of current imaging concepts and application, with the next generation and emerging Inverse Scattering Series imaging concepts and methods.