  Starting from a summary of detection statistics of our recent X-shooter
campaign, we review the major surveys, both space and ground based, for
emission counterparts of high-redshift damped Ly$\alpha$ absorbers (DLAs)
carried out since the first detection 25 years ago. We show that the detection
rates of all surveys are precisely reproduced by a simple model in which the
metallicity and luminosity of the galaxy associated to the DLA follow a
relation of the form, ${\rm M_{UV}} = -5 \times \left(\,[{\rm M/H}] + 0.3\,
\right) - 20.8$, and the DLA cross-section follows a relation of the form
$\sigma_{DLA} \propto L^{0.8}$. Specifically, our spectroscopic campaign
consists of 11 DLAs preselected based on their equivalent width of SiII
$\lambda1526$ to have a metallicity higher than [Si/H] > -1. The targets have
been observed with the X-shooter spectrograph at the Very Large Telescope to
search for emission lines around the quasars. We observe a high detection rate
of 64% (7/11), significantly higher than the typical $\sim$10% for random,
HI-selected DLA samples. We use the aforementioned model, to simulate the
results of our survey together with a range of previous surveys: spectral
stacking, direct imaging (using the `double DLA' technique), long-slit
spectroscopy, and integral field spectroscopy. Based on our model results, we
are able to reconcile all results. Some tension is observed between model and
data when looking at predictions of Ly$\alpha$ emission for individual targets.
However, the object to object variations are most likely a result of the
significant scatter in the underlying scaling relations as well as
uncertainties in the amount of dust which affects the emission.
