We have recently hit the milestone of 5,000 exoplanets discovered. In stark contrast with the Solar System, most of the exoplanets we know to date orbit extremely close to their host stars, causing them to lose copious amounts of gas through atmospheric escape at some stage in their lives. In some planets, this process can be so dramatic that they shrink in timescales of a few million to billions of years, imprinting features in the demographics of transiting exoplanets. Depending on the transit geometry, ionizing conditions, and atmospheric properties, a planetary outflow can be observed using transmission spectroscopy in the ultraviolet, optical or near-infrared. In this review, we will discuss the main techniques to observe evaporating exoplanets and their results. To date, we have evidence that at least 28 exoplanets are currently losing their atmospheres, and the literature has reported at least 42 non-detections.