Quasielastic neutron scattering of normal and supercooled water was measured in the temperature range 253-330K to investigate the possible anomaly expected to occur around 230K. Anomalous temperature dependence of self-diffusion coefficient, which may suggest a critical slowing-down, was found for the first time. Vapor-deposited glassy water (amorphous ice) and those doped with methanol (hydrogen-bond breaking impurity) and Xe and SF6 (both hydrogen-bond forming impurities) were prepared in a novel cryostat designed for this experiment. The in-situ neutron scattering of these samples was measured on a total scattering, smallangle scattering, and inelastic scattering spectrometers. A strong correlation among the ordering of hydrogenbond system, volume expansion, homogenization in a nano-scale range, and decrease of the excess lowenergy excitation was found in the experiments for the annealing and impurity effects. The future study on supercooled and glassy water was discussed along with high-intensity spectrometers at J-PARC.