Most planets are surrounded by a layer of material in gas phase with varying levels of physical and chemical properties. The planetary upper atmosphere invariably extend to space with decreasing levels of density and increasing gas particle velocities the higher we are in altitude. Some of these particles are so fast that they can unbind from the gravitational pull of the planet, and eventually escape it. Although seemingly mundane in our lives, this process of evaporation is precisely what transformed Mars from a world covered by rivers and oceans into a desolate, uninhabitable planet. More than 4,000 exoplanets have been discovered to date, and most of them orbit closer to their host star than Mercury orbits the Sun. Their extreme environments drive atmospheric escape rates high enough to strip them of a large fraction of their primordial envelopes, and we have the tools to observe this process happening. In this manuscript, I will describe in detail the state-of-the-art hypothesis and theories to explain atmospheric escape in exoplanets, as well as the leading-edge techniques to observe it. We shall embark on a journey from gas giants to Earth twins, in which we search for the faintest signatures of their evolution imprinted in their thin, upper atmospheres. To this end, we employ transmission spectroscopy, a technique that consists in measuring the light of their host stars filtered through the thin gaseous limbs of a transiting exoplanet, and inferring the properties of their atmospheres. Our observations are carried out in ultraviolet wavelengths using the Hubble Space Telescope and in infrared using ground-based telescopes. Leveraging what we learned in this journey, we shall also explore the possibility of probing the presence of water and habitability of Earth-like exoplanets using the future LUVOIR mission.