  We present the full results of our decade-long astrometric monitoring
programs targeting 31 ultracool binaries with component spectral types M7-T5.
Joint analysis of resolved imaging from Keck Observatory and Hubble Space
Telescope and unresolved astrometry from CFHT/WIRCam yields parallactic
distances for all systems, robust orbit determinations for 23 systems, and
photocenter orbits for 19 systems. As a result, we measure 38 precise
individual masses spanning 30-115 $M_{\rm Jup}$. We determine a
model-independent substellar boundary that is $\approx$70 $M_{\rm Jup}$ in mass
($\approx$L4 in spectral type), and we validate Baraffe et al. (2015)
evolutionary model predictions for the lithium-depletion boundary (60 $M_{\rm
Jup}$ at field ages). Assuming each binary is coeval, we test models of the
substellar mass-luminosity relation and find that in the L/T transition, only
the Saumon & Marley (2008) "hybrid" models accounting for cloud clearing match
our data. We derive a precise, mass-calibrated spectral type-effective
temperature relation covering 1100-2800 K. Our masses enable a novel direct
determination of the age distribution of field brown dwarfs spanning L4-T5 and
30-70 $M_{\rm Jup}$. We determine a median age of 1.3 Gyr, and our population
synthesis modeling indicates our sample is consistent with a constant star
formation history modulated by dynamical heating in the Galactic disk. We
discover two triple-brown-dwarf systems, the first with directly measured
masses and eccentricities. We examine the eccentricity distribution, carefully
considering biases and completeness, and find that low-eccentricity orbits are
significantly more common among ultracool binaries than solar-type binaries,
possibly indicating the early influence of long-lived dissipative gas disks.
Overall, this work represents a major advance in the empirical view of very
low-mass stars and brown dwarfs.
