  Quantifying the physical conditions that allow radiation emitted shortward of
the hydrogen ionization edge at 911.7 {\AA} to escape the first collapsed
objects and ultimately reionize the universe is a compelling problem for
astrophysics. The escape of LyC emission from star-forming galaxies and AGN is
intimately tied to the emergence and sustenance of the metagalactic ionizing
background that pervades the universe to the present day and in turn is tied to
the emergence of structure at all epochs. JWST was built in part to search for
the source(s) responsible for reionization, but it cannot observe LyC escape
directly, because of the progressive increase in the mean transmission of the
intergalactic medium towards the epoch of reionization. Remarkable progress has
been made to date in directly detecting LyC leaking from star-forming galaxies
using space-based and the ground-based observatories, but there remain
significant gaps in our redshift coverage of the phenomenon. Ongoing projects
to measure LyC escape at low- and intermediate-z will provide guidance to JWST
investigations by analyzing the robustness of a set of proposed LyC escape
proxies, and also provide a closeup examination of the physical conditions that
favor LyC escape. However, currently available facilities are inadequate for
deeply probing LyC escape at the faint end of the galaxy luminosity function.
Doing so will require facilities that can detect LyC emission in the restframe
to limiting magnitudes approaching 28 $< m^*_{(1+z)900} <$ 32 for
$M^*_{(1+z)1500}$ galaxies. The goal of acquiring statistically robust samples
for determining LyC luminosity functions across cosmic time will require
multi-object spectroscopy from spacebased flagship class and groundbased ELT
class telescopes along with ancillary panchromatic imaging and spectroscopy
spanning the far-UV to the mid-IR.
