Primary auditory cortex is a critical stage in the human auditory pathway, a gateway between subcortical and higher-level cortical areas. Receiving the output of all subcortical processing, it sends its output on to higher-level cortex. Non-invasive physiological recordings of primary auditory cortex using electroencephalography (EEG) and magnetoencephalography (MEG), however, may not have sufficient specificity to separate responses generated in primary auditory cortex from those generated in underlying subcortical areas or neighboring cortical areas. This limitation is important for investigations of effects of top-down processing (e.g., selective-attention-based) on primary auditory cortex: higher-level areas are known to be strongly influenced by top-down processes, but subcortical areas are often assumed to perform strictly bottom-up processing. Fortunately, recent advances have made it easier to isolate the neural activity of primary auditory cortex from other areas. In this perspective, we focus on time-locked responses to stimulus features in the high gamma band (70-150 Hz) and with early cortical latency (similar to 40 ms), intermediate between subcortical and higher-level areas. We review recent findings from physiological studies employing either repeated simple sounds or continuous speech, obtaining either a frequency following response (FFR) or temporal response function (TRF). The potential roles of top-down processing are underscored, and comparisons with invasive intracranial EEG (iEEG) and animal model recordings are made. We argue that MEG studies employing continuous speech stimuli may offer particular benefits, in that only a few minutes of speech generates robust high gamma responses from bilateral primary auditory cortex, and without measurable interference from subcortical or higher-level areas.