While the water molecule is simple, its condensed-phase liquid behavior is so complex that no consensus description has emerged despite three centuries of effort. Here we identify features of its behavior that are the most peculiar, hence suggest ways forward. We examine the properties of water at the boundaries of common experience, including stable states at high pressure, the supercooled state at normal and elevated pressure, and the stretched (“negative pressure”) state out to the limits of mechanical stability. The familiar anomalies at moderate pressures (viscosity and density (TMD) behavior, etc.), are not explained by H-bond breaking, according to common bond-breaking criteria. A comparison of data on the TMD, at both positive and negative pressures, with the predictions of popular pair potential models, shows dramatic discrepancies appearing in the stretched liquid domain. This prompts questions on the second-critical-point (TC2) hypothesis that has been guiding much current thinking. We turn to related systems for guidance, reviewing a hierarchy of water-like anomalies. We conclude that water models are far from complete and that proper understanding of water will depend on the success in mastering the measurement of liquid behavior in the negative-pressure domain —which we discuss.