We investigate the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquid‐liquid critical point in (i) Two models of water, TIP5P and ST2, which display a first order liquid‐liquid phase transition at low temperatures; (ii) the Jagla model, a spherically symmetric two‐scale potential known to possess a liquid‐liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions; and (iii) A Hamiltonian model of water where the idea of two length/energy scales is built in; this model also displays a first order liquid‐liquid phase transition at low temperatures besides the first order liquid‐gas phase transition at high temperatures. We find a correlation between the dynamic fragility crossover and the locus of specific heat maxima C P max (“Widom line”) emanating from the critical point. Our findings are consistent with a possible relation between the previously hypothesized liquid‐liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water. More generally, we argue that this connection between C P max and the dynamic crossover is not limited to the case of water, a hydrogen bonded network liquid, but is a more general feature of crossing the Widom line, an extension of the first‐order coexistence line in the supercritical region. We present evidence from experiments and computer simulations supporting the hypothesis that water displays polyamorphism, i.e., water separates into two distinct liquid phases. This concept of a new liquid‐liquid phase transition is finding application to other liquids as well as water, such as silicon and silica. We also discuss related puzzles, such as the mysterious behavior of confined water and the “skin” of hydration water near a biomolecule. Specifically, using molecular dynamics simulations, we also investigate the relation between the dynamic transitions of biomolecules (lysozyme and DNA) and the dynamic and thermodynamic properties of hydration water. We find that the dynamic transition of the macromolecules, sometimes called a “protein glass transition”, occurs at the temperature of dynamic crossover in the diffusivity of hydration water, and also coincides with the maxima of the isobaric specific heat C P and the temperature derivative of the orientational order parameter. We relate these findings to the hypothesis of a liquid‐liquid critical point in water. Our simulations are consistent with the possibility that the protein glass transition results from a change in the behavior of hydration water, specifically from crossing the Widom line.