Water is a fascinating material. Its composition is simple—one oxygen and two hydrogen atoms—but its chemistry and physics are extremely complex and exhibit 75 documented anomalies. Although these anomalies and their molecular origin are not completely understood, we know that hydrogen bonds play key roles in all of the phases of water. Moreover, there is experimental evidence that the polymorphism of the ice structure extends into the liquid phase and is associated with a liquid-liquid coexistence line. This is currently a topic of great interest in water research because there are indications that the end point of the coexistence line corresponds to a second critical point inside the supercooled liquid regime. We examine the recent progress in understanding water anomalies and the liquid-liquid phase transition hypothesis, including the results of recent experimental work and molecular simulations of both bulk and confined water. We examine experimental results that test whether the behavior of liquid water is consistent with the “liquid polymorphism” hypothesis that liquid water can exist in two distinct phases of differing densities. We also examine recent research on the anomalies of nanoconfined water and, in particular, on water in biological environments. We find that the concept of liquid polymorphism can also describe the properties of other liquids that have two characteristic length scales.