We review the recent research on supercooled and glassy water, focusing
on the possible origins of its complex behavior. We stress the central
role played by the strong directionality of the water-water interaction
and by the competition between local energy, local entropy, and local
density. In this context we discuss the phenomenon of polyamorphism
(i.e., the existence of more than one disordered solid state),
emphasizing both the role of the preparation protocols and the
transformation between the different disordered ices. Finally, we
present the ongoing debate on the possibility of linking polyamorphism
with a liquid-liquid transition that could take place in the no-man's
land, the temperature-pressure window in which homogeneous nucleation
prevents the investigation of water in its metastable liquid form
