Pumping energy is supplied by temperature changes alone. A general feature of such pumps is that the upper pressure limit is reached when the mean free path becomes small relative to the physical dimensions of the pump in the region of the temperature transition. Thus, the upper pressure limits of these pumps have been determined by the microfabrication limits of the day; they have operated at relatively low pressures, with low throughputs, and have not become main line pumps. In recent years, however, Micro-Electronic-Mechanical Systems (MEMS) has introduced a whole new level of miniaturization to devices in general, including vacuum devices, and hence has raised the upper pressure limits, and thus the throughputs of thermal molecular pumps to atmospheric levels. The purpose of this article is to isolate the various physical manifestations of thermal molecular Dumps, which have been realized over the years. The general pumping phenomenon has had various names: Knudsen compressor; thermal transpiration; thermal creep; thermodynamic, thermolecular, thermal molecular, and accommodation pumping. This multiplicity of terminology can cause some confusion and it is one of the aims of the article to simplify the situation. We have chosen the title "Pumping by Thermal Molecular Pressure" following the terminology of Knudsen. Broadly speaking, it is found that these pumps divide into two classes: (a) those requiring no specially prepared surfaces, (b) those in which special surfaces are essential. The latter have no low pressure limit. A table is assembled comparing pumps which have been built and tested, rather than those calculated on paper. Scaling rules for multiple stage pumps in series, based on results obtained for single-stage pumps are presented. A Knudsen compressor in series with an accommodation pump already promises operation from atmosphere to indefinitely low pressures, whereas an accommodation pump alone may be able to cover this range in the future. A number of potential applications of the technology such as small gas chromatographs and small valves are noted. Despite this complexity, thermal molecular pressure devices all have the compelling advantage that there are no moving parts nor any fluids in the vacuum.