Water is one of the most abundant molecules on Earth, of paramount importance to our daily lives and is of great relevance in astrophysics. Nevertheless its physical and chemical properties, which are often called anomalous, are not fully understood by now. Investigations in recent decades have shown that water exists in many crystalline forms — a phenomenon known as “polymorphism” — and in three amorphous forms — a phenomenon known as “polyamorphism”. In this article we review the crystalline ice phases and outline possibilities for future experimental discoveries of ice polymorphs. We then provide an overview about the current knowledge on polyamorphism and finally go into more detail about the question whether or not the amorphous ices are linked by glass-to-liquid transitions to deeply supercooled liquids, which has been a major focus in our research group over the last years.