The idea that a thermodynamic glass transition of some sort underlies the observedglass formation has been highly debated since Kauzmann first stressed the hypothet-ical entropy crisis that could take place if one were able to equilibrate supercooledliquids below the experimental glass transition temperature Tg . This a priori un-reachable transition at some TK < Tg has since received a firm theoretical basis as akey feature predicted by the mean-field theory of the glass transition. In this chapter,we assess whether, and in which form, such a transition can survive in finite dimen-sions, and we review some of the recent computer simulation work addressing theissue in 2- and 3-dimensional glass-forming liquid models. We also discuss theoreticalreasons to focus on an apparently inaccessible singularity.