This volume of Advances in Chemical Physics contains a representative cross-section of research that questions established thinking by scrutinizing the evidence for and against, and inquiring into the possibility and consequences of, the existence of more than one liquid phase in a pure substance. Chapters address Electron Spin Resonance Studies of Supercooled Water; Pressure-Driven Liquid-Liquid Transformations and Corresponding Bizarre Viscosity Behavior; The Stability Limit and Other Open Questions on Water at Negative Pressure; Water-like Anomalies of Core-Softened Fluids: Dependence on the Trajectories in ( Pρ T) Space; High-Frequency Dynamics of Liquids through a Liquid-Liquid Transition: The Case of Cs; "The Liquid-Liquid Phase Transition, Anomalous Properties, and Glass Behavior of Polymorphic Liquids; Amorphous Ices; Water Proton Environment: A New Water Anomaly at Atomic Scale?; Polymorphism and Anomalous Melting in Isotropic Fluids; "Transport and Dynamics in Supercooled Confined Water; Water and Biological Macromolecules; "Polyamorphism and Liquid-Liquid Phase Transitions in Amorphous Silicon and Supercooled Al2O3-Y2O3 Liquids; Polyamorphism in Water; "Computer Simulations of Liquid Silica: Water-Like Thermodynamic and Dynamic Anomalies, and the Evidence for Polyamorphism; "Polymorphism in Lattice Models; Cooperative Bond Ordering in Liquid: Its Link to Liquid Polymorphism and Water-Like Anomalies; Statistical Mechanical Approach to the Thermodynamic Stability of Clathrate Hydrates; Liquid-Liquid Phase Transition in Supercooled Silicon; and Similarities of the Collective Interfacial Dynamics of Grain Boundaries and Nanoparticles to Glass-Forming Liquids.