  This is a survey covering aspects of varied work of the authors with Mohammed
Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally
canonical sections of ample line bundles on abelian varieties, we motivate,
using mirror symmetry, the idea that theta functions exist in much greater
generality. This suggestion originates with the work of the late Andrei Tyurin.
We outline how to construct theta functions on the degenerations of varieties
constructed in previous work of the authors, and then explain applications of
this construction to homological mirror symmetry and constructions of broad
classes of mirror varieties.
