  We present some of the results of a large survey aimed at establishing the
properties of galaxies selected by their having produced detectable Mg~II
$\lambda\lambda 2796$, 2803 absorption in the spectra of background QSOs. The
present sample covers the redshift range $0.2 \le z \le 1.0 $, with $\langle z
\rangle = 0.65$. From an extensive program of optical and near-IR imaging and
optical spectroscopy, we find that the galaxies appear to be similar to normal
galaxies at the present epoch, ranging from late--type spiral galaxies to those
whose spectra and colors resemble present--day ellipticals. Contrary to some
faint field galaxy samples selected using different criteria, over the redshift
range observed we find no evidence for significant evolution in rest--frame
$B-K$ color, space density, or (rest--frame $B$ or $K$) luminosity. The
``average'' Mg~II absorbing galaxy appears to be consistent with a normal
0.7$L_B^{\ast}$ Sb galaxy having a roughly constant star formation rate since
$z \sim 1$, although galaxies spanning a range of a factor of $\sim 70$ in
luminosity are found in the absorber sample. The diffuse gas cross-section
selection imposed by studies of this kind appears to be biased {\it against}
the relatively underluminous, blue galaxies which apparently dominate the
number counts at faint magnitudes. However, essentially all ``normal'' field
galaxies, independent of spectroscopic type, appear to be potential QSO
absorbers.
