  The unitarity of the lepton mixing matrix is a critical assumption underlying
the standard neutrino-mixing paradigm. However, many models seeking to explain
the as-yet-unknown origin of neutrino masses predict deviations from unitarity
in the mixing of the active neutrino states. Motivated by the prospect that
future experiments may provide a precise measurement of the lepton mixing
matrix, we revisit current constraints on unitarity violation from oscillation
measurements and project how next-generation experiments will improve our
current knowledge. With the next-generation data, the normalizations of all
rows and columns of the lepton mixing matrix will be constrained to
$\lesssim$10\% precision, with the $e$-row best measured at $\lesssim$1\% and
the $\tau$-row worst measured at ${\sim}10\%$ precision. The measurements of
the mixing matrix elements themselves will be improved on average by a factor
of $3$. We highlight the complementarity of DUNE, T2HK, JUNO, and IceCube
Upgrade for these improvements, as well as the importance of $\nu_\tau$
appearance measurements and sterile neutrino searches for tests of leptonic
unitarity.
