  From as early as the 1930s, astronomers have tried to quantify the
statistical nature of the evolution and large-scale structure of galaxies by
studying their luminosity distribution as a function of redshift - known as the
galaxy luminosity function (LF). Accurately constructing the LF remains a
popular and yet tricky pursuit in modern observational cosmology where the
presence of observational selection effects due to e.g. detection thresholds in
apparent magnitude, colour, surface brightness or some combination thereof can
render any given galaxy survey incomplete and thus introduce bias into the LF.
  Over the last seventy years there have been numerous sophisticated
statistical approaches devised to tackle these issues; all have advantages --
but not one is perfect. This review takes a broad historical look at the key
statistical tools that have been developed over this period, discussing their
relative merits and highlighting any significant extensions and modifications.
In addition, the more generalised methods that have emerged within the last few
years are examined. These methods propose a more rigorous statistical framework
within which to determine the LF compared to some of the more traditional
methods. I also look at how photometric redshift estimations are being
incorporated into the LF methodology as well as considering the construction of
bivariate LFs. Finally, I review the ongoing development of completeness
estimators which test some of the fundamental assumptions going into LF
estimators and can be powerful probes of any residual systematic effects
inherent magnitude-redshift data.
