Leonid Keldysh’s 1965 paper "Ionization in the Field of a Strong Electromagnetic Wave" contains useful and simple-to-implement equations predicting the rate of photoionization in both atomic gasses and general crystalline solids. Crucially, his result for the case of crystalline solids does not require that the wave functions of the electron bands be specified. However, the derivation of his method is not presented in detail in his original paper, nor has it been in other publications. This presents difficulty for anyone wishing to understand his approach or adapt it to better suit their own needs. This thesis provides a more involved description of his method for the atomic case, and particular attention is given to the crystalline case. The basic structure of his method for both cases is described in detail, and the final rate equation and asymptotic approximations are produced for the case of crystalline solids.