The history of science, in particular the field of physics, contains examples of a theoretical development preceding its experimental verification. This situation repeatedly occurred in nonlinear spectroscopy, as the experimental measurement of electronic transitions that occurs in nonlinear processes required high-intensity sources of radiation. In this chapter I describe the long developmental path from the 1929 publication of Maria GA¶ppert-Mayer on the theory of two-photon absorption and emission in atoms to the 1990 publication of Denk, Strickler, and Webb that demonstrated two-photon microscopy. The bridge between the work of GA¶ppert-Mayer and the experimental realization in 1990 was the work and publications of the group at Oxford working on nonlinear scanning optical microscopy. 11.1 GA¶ppert-Mayer's Theory of Two-Photon Absorption The theoretical basis for two-photon quantum transitions (absorption and emission) in atoms was the subject of a doctoral thesis published in 1931 by Maria GA¶ppert-Mayer (see Fig. 11.1). Two years earlier she published a preliminary paper on her theory (GA¶ppert-Mayer, 1929), in which she formulated energy-state diagrams for both two-photon emission and two-photon absorption processes. She indicated the presence of virtual states, and she concluded that the probability for the two-photon absorption process is proportional to the square of the light intensity. In her 1931 dissertation, GA¶ppert-Mayer followed the technique of Dirac for the use of perturbation theory to solve the quantum-mechanical equations for the processes of absorption, emission, and dispersion of light in single photon-atom interactions. The transition probability of a two-photon electronic process was derived by using second-order, time-dependent perturbation theory. Her derivation clearly states that the probability of a two-photon absorption process is quadratically related to the excitation light intensity. For readers who cannot read the 1931 dissertation in German, I have made a translation into English. This translation will appear as a chapter in the new book to be published by Oxford University Press: Handbook of Biological Nonlinear Microscopy (Masters, So, 2006). An important aspect of GA¶ppert-Mayer's work is that the process of two-photon absorption involves the interaction of two photons and an atom. This interaction must occur within the lifetime of an intermediate virtual state, which can be described as a superposition of states and not an eigenstate of the atom.