This entry is about the notion of “crystal” in algebraic geometry. For the notion in solid state physics see at crystal. There are few mutually unrelated notions denoted by “crystal” in mathematics. One can of course talk about mathematical models of physical crystals and their geometry. Another, is an intermediary notion leading to crystal bases of Kashiwara and of Lusztig, thus one associates a crystal to a quantized enveloping algebra. Finally, there are crystals due Grothendieck to which...