  Parallaxes, proper motions, and optical photometry are presented for 51
systems made up 37 cool subdwarf and 14 additional high proper motion systems.
Thirty-seven systems have parallaxes reported for the first time, 15 of which
have proper motions of at least 1"/yr. The sample includes 22 newly identified
cool subdwarfs within 100 pc, of which three are within 25 pc, and an
additional five subdwarfs from 100-160 pc. Two systems --- LSR 1610-0040 AB and
LHS 440 AB --- are close binaries exhibiting clear astrometric perturbations
that will ultimately provide important masses for cool subdwarfs.
  We use the accurate parallaxes and proper motions provided here, combined
with additional data from our program and others to determine that effectively
all nearby stars with tangential velocities greater than 200 km s$^{-1}$ are
subdwarfs. We compare a sample of 167 confirmed cool subdwarfs to nearby main
sequence dwarfs and Pleiades members on an observational Hertzsprung-Russell
diagram using $M_V$ vs.~$(V-K_{s})$ to map trends of age and metallicity. We
find that subdwarfs are clearly separated for spectral types K5--M5, indicating
that the low metallicities of subdwarfs set them apart in the H-R diagram for
$(V-K_{s})$ = 3--6. We then apply the tangential velocity cutoff and the
subdwarf region of the H-R diagram to stars with parallaxes from {\it Gaia}
Data Release 1 and the MEarth Project to identify a total of 29 new nearby
subdwarf candidates that fall clearly below the main sequence.
