Exoplanets with substantial hydrogen/helium atmospheres have been discovered in abundance, many residing extremely close to their parent stars. The extreme irradiation levels that these atmospheres experience cause them to undergo hydrodynamic atmospheric escape. Ongoing atmospheric escape has been observed to be occurring in a few nearby exoplanet systems through transit spectroscopy both for hot Jupiters and for lower-mass super-Earths and mini-Neptunes. Detailed hydrodynamic calculations that incorporate radiative transfer and ionization chemistry are now common in one-dimensional models, and multidimensional calculations that incorporate magnetic fields and interactions with the interstellar environment are cutting edge. However, comparison between simulations and observations remains very limited. While hot Jupiters experience atmospheric escape, the mass-loss rates are not high enough to affect their evolution. However, for lower-mass planets, atmospheric escape drives and controls their evolution, sculpting the exoplanet population that we observe today. Observations of some exoplanets have detected atmospheric escape driven by hydrodynamic outflows, causing the exoplanets to lose mass over time. Hydrodynamic simulations of atmospheric escape are approaching the sophistication required to compare them directly to observations. Atmospheric escape sculpts sharp features into the exoplanet population that we can observe today; these features have recently been detected.