Band Theory has contributed immensely to our society. Without it we would never have understood semiconductors. Lacking that, the Information Revolution we are presently enjoying would never have occurred. However, is Band Theory sufficient to satisfactorily describe the electronic structure as well as the excitation spectra of all crystalline solids? Do other potential technological revolutions depend on the answer to the primary question? We will restrict our discussion of this question to results from photoemission spectroscopy (PES). PES was formally developed as an experimental technique starting about 1957. However, one can look upon Einstein's and Millikan's photoemission evidence for the photon as the first use of PES. A key physical problem in all photoemission is momentum conservation. The excited electron usually increases its momentum by much more than the momentum lost by the photon it absorbs; thus, there must be coupling to a third body (the crystalline lattice) which recoils in the excitation and in this way conserves momentum. a considerable portion of this paper is dedicated to examining the band theory question and its relation to the nature of the spatial and electronic structure of the photoemitting solid. Examples are given of cases where Band Theory may be insufficient to describe the excitation spectra and, perhaps, electron structure of large classes of crystalline materials. This paper is arranged roughly in a historic manner. It first examines the Einstein's photoemission predictions and shows why for almost four decades this early work established photoemission as only a surface phenomena. The work leading to the realization that bulk photoemission is usually dominant rather than surface photoemission is sketched out. Included is a section showing that if the dominant optical selection rules of Band Theory had applied, Einstein would have been proven wrong in his photoemission predictions. The development of what we now formally call photoemission spectroscopy (PES) starting in the late fifties is outlined. One element in this is the early attempts to use PES to test the limits of Band Theory. Finally, the PES results from a most interesting high temperature superconductor are examined. Here, the question of band theory is central and will be examined using PES studies which are among the most advanced.