Polyamorphism, i.e. the presence of more than one amorphous state, was observed for the first time in amorphous ice or glassy water. In addition to LDA (low-density amorphous ice), a second amorphous state, HDA (high-density amorphous ice), was discovered similar to 20 years ago. Since then, polyamorphism has been observed in many other substances, such as SiO2, GeO2, Si, and Ge. Five years ago, experimental results suggesting the existence of a third amorphous state, VHDA (very high-density amorphous ice), were reported, opening the possibility that more than two amorphous states could also be observed in other substances. A consistent phase diagram of glassy water does not yet exist. Such a phase diagram is necessary if one also wants to understand the anomalous behaviour of supercooled liquid water. Since the discovery of HDA, a large amount of work based on experiments and computer simulations has appeared. It is the purpose of this work to review such studies with special emphasis in comparing the experimental and simulation results. In particular, we review the recent studies concerning VHDA and its nature, and discuss the main open questions relating to the phase diagram of glassy water.