  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.
