This work will unite the closely related concepts of automorphic integral and nonanalytic automorphic form. We study nonanalytic automorphic integrals (NAIs), functions on the complex upper half-plane which satisfy a fairly general transformation law but retain some characteristics of the earlier concepts. We limit ourselves to the discrete Hecke groups, but allow complex coweights and a quite general class of period functions. Much is based on work of Knopp [Kn94] and Maass [Ma64]. We will give examples using the so-called weight-changing operators, several of which appear here for the first time. The appropriate Riemann-Hecke-Bochner Correspondence will be stated in Chapter Four and proved in Chapter Five. A conjecture in [Kn83] will be answered in the negative in Chapter Four.