Explicit solution(s) to the Schottky problem(s)

The Schottky problem is the question of characterizing Jacobians of curves among all principally polarized abelian varieties, and is the most classical of the kind of problems asking for a characterization of the image of the period mapping within the suitable period domain. I will review what is known about the classical and less classical Schottky problems, and present an explicit weak solution (joint with H. Farkas and R. Salvati Manni) to the most classical formulation of the classical Schottky problem.