The field's founding work is William James's study of religious experience, published in 1902 from lectures given in Edinburgh. Its method was to collect first-hand accounts of conversion, mysticism and the divided self, to describe them carefully, and to argue that their value should be judged by their fruits rather than by their origins. It set the terms for everything after, and it is still read.
The middle of the century was thinner, as psychology turned away from subjective report. The revival came from two directions: measurement of religious variables in large survey populations, and neuroscience.
The neuroscientific strand produced the field's most instructive failure. A laboratory reported that applying weak, complex magnetic fields to the temporal lobes induced a sensed presence in many subjects, and the apparatus was widely reported as a device for producing religious experience. An independent group then ran the procedure double blind, with subjects unaware whether the field was on, and found that reports were predicted by the subject's suggestibility and expectation and not by the field. The original laboratory disputed the replication's parameters, and the disagreement has not been resolved.
That episode arrived shortly before psychology's wider reckoning with replication, which has fallen on this subfield as on others, and it is why claims about the neural basis of religious experience should be read with the sample size and the blinding in view.