  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 $\sim
2$~M$_\oplus$. The size of these hydrogen dominated atmospheres indicates the
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 ($\lesssim$ 2~M$_\oplus$)
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).
