Evidence for exocomets, icy bodies in extrasolar planetary systems, has rapidly increased over the past decade. Volatiles are detected through the gas that exocomets release as they collide and grind down within their natal belts, or as they sublimate once scattered inwards to the regions closest to their host star. Most detections are in young, 10 to a few 100 Myr-old systems that are undergoing the final stages of terrestrial planet formation. This opens the exciting possibility to study exocomets at the epoch of volatile delivery to the inner regions of planetary systems. Detection of molecular and atomic gas in exocometary belts allows us to estimate molecular ice abundances and overall elemental abundances, enabling comparison with the Solar Nebula and Solar System comets. At the same time, observing star-grazing exocomets transiting in front of their star (for planetary systems viewed edge-on) and exozodiacal dust in the systems innermost regions gives unique dynamical insights into the inward scattering process producing delivery to inner rocky planets. The rapid advances of this budding subfield of exoplanetary science will continue in the short term with the upcoming JWST, WFIRST and PLATO missions. In the longer term, the priority should be to explore the full composition of exocomets, including species crucial for delivery and later prebiotic synthesis. Doing so around an increasingly large population of exoplanetary systems is equally important, to enable comparative studies of young exocomets at the epoch of volatile delivery. We identify the proposed LUVOIR and Origins flagship missions as the most promising for a large-scale exploration of exocometary gas, a crucial component of the chemical heritage of young exo-Earths.