The discovery of thousands of highly irradiated, low-mass, exoplanets has led to the idea that atmospheric escape is an important process that can drive their evolution. Of particular interest is the inference from recent exoplanet detections that there is a large population of low mass planets possessing significant, hydrogen dominated atmospheres, even at masses as low as ∼2 M⊕. The size of these hydrogen dominated atmospheres indicates the envelopes must have been accreted from the natal protoplanetary disc. This inference is in contradiction with the Solar System terrestrial planets, that did not reach their final masses before disc dispersal, and only accreted thin hydrogen dominated atmospheres. In this review, we discuss the evidence for hydrogen dominated atmospheres on terrestrial mass (≲2 M⊕) planets. We then discuss the possible origins and evolution of these atmospheres with a focus on the role played by hydrodynamic atmospheric escape driven by the stellar high-energy emission (X-ray and EUV; XUV).