Intended for advanced undergraduates and graduate students in mathematics, physics, and chemistry, this work teaches problem-solving using the theory of special functions. The concise treatment presents the theory methodically and in detail to a wide variety of problems in atomic and molecular physics. The overall applicability of this method and its extension to solving these problems are discussed with attention to detail seldom found in textbooks of this level. Starting with a brief introduction to the hypergeometric equations and their properties, a step-by-step method consisting of six distinct parts illustrates how to address typical problems in quantum physics in a simple and uniform fashion. This technique can also be applied to the solution of other problems, for which the Schrödinger equation can be reduced by some means to an equation of the hypergeometric type. Topics include the discrete spectrum eigenfunctions, linear harmonic oscillators, Kratzer molecular potential, and the rotational correction to the Morse formula. The text concludes with an Appendix that presents an original Fourier transform-based method for converting multicenter integrals to a single center.