There has been uninterrupted interest in supercooled water ever since the pioneering experiments of Speedy and Angell revealed sharp increases in this substance's response functions upon supercooling. One intriguing hypothesis that was formulated to explain this behavior involves the existence of a metastable liquid-liquid transition (LLT) at deeply supercooled conditions. The preponderance of experimental evidence is consistent with this hypothesis, although no definitive proof exists to date. Computational studies have played an important role in this area, because ice nucleation can in principle be controlled in silico. It has been claimed, controversially, that the LLT is a misinterpreted liquid-solid transition in all models of water. Recent studies disprove this viewpoint by providing unambiguous counter-examples of distinct liquid-liquid and liquid-crystal transitions in tetrahedral models. In one, state-of-the-art sampling methods were used to compute the free energy surface of a molecular model of water and revealed the existence of two liquid phases in metastable equilibrium with each other and a stable crystal phase, at the same, deeply supercooled thermodynamic conditions. Further studies showed that, by tuning the potential parameters of a model tetrahedral system, it is possible to make the LLT evolve continuously from metastability to being thermodynamically stable with respect to crystallization. Most recently, it has been shown that the simulation code used to challenge the hypothesis of an LLT contains conceptual errors that invalidate the results on which the challenge was based, definitively resolving the controversy. The debate has vastly expanded the range of fundamental questions being pursued about phase transitions in metastable systems and ushered the use of increasingly sophisticated computational methods to explore the possible existence of LLTs in model systems.