  We have used the Atacama Large Millimeter/submillimeter Array (ALMA) to carry
out a search for CO (3$-$2) or (4$-$3) emission from the fields of 12
high-metallicity ([M/H]~$\geq -0.72$\,dex) damped Lyman-$\alpha$ absorbers
(DLAs) at $z \approx 1.7-2.6$. We detected CO emission from galaxies in the
fields of five DLAs (two of which have been reported earlier), obtaining high
molecular gas masses, $\rm M_{mol} \approx (1.3 - 20.7) \times (\alpha_{\rm
CO}/4.36) \times 10^{10} \; M_\odot$. The impact parameters of the CO emitters
to the QSO sightline lie in the range $b \approx 5.6-100$~kpc, with the three
new CO detections having $b \lesssim 15$~kpc. The highest CO line luminosities
and inferred molecular gas masses are associated with the highest-metallicity
DLAs, with [M/H]~$\gtrsim -0.3$\,dex. The high inferred molecular gas masses
may be explained by a combination of a stellar mass-metallicity relation and a
high molecular gas-to-stars mass ratio in high-redshift galaxies; the DLA
galaxies identified by our CO searches have properties consistent with those of
emission-selected samples. None of the DLA galaxies detected in CO emission
were identified in earlier optical or near-IR searches and vice-versa; DLA
galaxies earlier identified in optical/near-IR searches were not detected in CO
emission. The high ALMA CO and C[{\sc ii}]~158$\mu$m detection rate in
high-$z$, high-metallicity DLA galaxies has revolutionized the field, allowing
the identification of dusty, massive galaxies associated with high-$z$ DLAs.
The H{\sc i}-absorption criterion identifying DLAs selects the entire high-$z$
galaxy population, including dusty and UV-bright galaxies, in a wide range of
environments.
