The conclusive observation of a second-order chiral phase transition in QCD has eluded both theory and experiment so far. We expect such a transition at lighter than physical quark masses and possibly also at finite density. In the vicinity of second-order phase transitions universal critical phenomena yield powerful constraints for the system. In addition, information about the phase structure is encoded in the analytic structure of thermodynamic quantities. I will discuss how this information can be leveraged to understand the QCD phase transition a bit better. Of central interest are branch points of the free energy, known as Yang-Lee edge singularities. They are in general located, e.g., at complex quark masses and chemical potentials, but pinch the real axes where second-order transitions occur. This information can be used to understand the nature of the critical mode in hot and dense QCD. I will also show how further analytic constraints on the location of the edge singularities in the complex plane can be exploited to learn about the dependence of the chiral phase transition on the quark masses.