There has been considerable effort expended by theorists to explain the occurrence and nature of high temperature superconductivity (HTS) in oxide systems. Although it is generally believed that electron pairing is consistent with all of the measurements, there is no consensus on the nature of the pairing or the microscopic mechanism for HTS.Two areas of focus are the mechanism for binding the pairs and the characterization of the paired state. It has been argued that the pairing boson can be a phonon as in the standard BCS theory and its refinements. However, objections have been raised about various models based on phonons. Other proposals suggest using plasmons, excitons, demons, and magnons for pairing. Many of the models proposed have attractive features, but again, there are shortcomings or insufficient experimental support. Some suggestions such as the RVB model claim that the normal state is not a Fermi liquid and that the standard approaches are not applicable. This has provided added motivation for studying the normal state in more detail.Because of the complexity of the oxide superconductors and the observation of strong phonon, electronic, and spin effects, it is still difficult to provide verdicts for the proposed pairing mechanisms except in those cases where detailed calculations have been done. The expected short coherence lengths put restrictions on the nature of the pairs. Because of this, HTS theories range from BCS pairing to Bose-Einstein condensation.An overview consisting of some history, background, and a discussion of some specific theoretical proposals is given here. Some suggestions are made for using experiment to reduce the number of promising theoretical choices. For non-oxide systems, a discussion is given of the possibility of phonon-induced superconductivity in metallic hydrogen in the range of 230 K.