In this talk I will discuss the Schottky problem determining which complex principally polarized abelian varieties arise as Jacobians of smooth algebraic curves. Theta functions play a pivotal role in the study of the Schottky problem.
In the first part of my talk I briefly survey the historical background of the Schottky problem. In the second part of my talk, I will discuss the recent progress towards the Schottky problem. Finally I will make some comments on the relations among the Schottky problem, toroidal compactifications of the Siegel space and the Siegel-Jacobi space, the André-Oort conjecture, Okounkov bodies and the Coleman's conjecture, and then provide some open problems which lead to a new approach towards the study of the Schottky problem.