After providing some background material to establish the interest content of this subject, we summarize the many different ways in which water can be prepared in the amorphous state, making clear that there seems to be more than one distinct amorphous state to be considered. We then give some space to structural and spectroscopic characterization of the distinct states, recognizing that whereas there seems to be unambiguously two distinct states, there may be in fact be more, the additional states mimicking the structures of the higher-density crystalline polymorphs. The low-frequency vibrational properties of the amorphous solid states are then examined in some detail because of the gathering evidence that glassy water, while difficult to form directly from the liquid like other glasses, may have some unusual and almost ideal glassy features, manifested by unusually low states of disorder. This notion is pursued in the following section dealing with thermodynamic and relaxational properties, where the uniquely low excess entropy of the vitreous state of water is confirmed by three different estimates. The fact that the most nearly ideal glass known has no properly established glass transition temperature is highlighted, using known dielectric loss data for amorphous solid water (ASW) and relevant molecular glasses. Finally, the polyamorphism of glassy water, and the kinetic aspects of transformation from one form to the other, are reviewed.