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The thermal restoration of chiral symmetry in QCD is known to proceed by ananalytic crossover, which is widely expected to turn into a phase transitionwith a critical endpoint as the baryon density is increased. In the absence ofa genuine solution to the sign problem of lattice QCD, simulations at zero andimaginary baryon chemical potential in a parameter space enlarged by a variablenumber of quark flavours and quark masses constitute a viable way to constrainthe location of a possible non-analytic phase transition and its criticalendpoint. In this article I review recent progress towards an understanding ofthe nature of the transition in the massless limit, and its criticaltemperature at zero density. Combined with increasingly detailed studies of thephysical crossover region, current data bound a possible critical point to$\mu_B > 3T$.

Lattice Constraints on the QCD Chiral Phase Transition at Finite Temperature and Baryon Density