  Novikov's conjecture on the Riemann-Schottky problem: {\it the Jacobians of
smooth algebraic curves are precisely those indecomposable principally
polarized abelian varieties (ppavs) whose theta-functions provide solutions to
the Kadomtsev-Petviashvili (KP) equation}, was the first evidence of nowadays
well-established fact: connections between the algebraic geometry and the
modern theory of integrable systems is beneficial for both sides.
  The purpose of this paper is twofold. Our first goal is to present a proof of
the strongest known characterization of a Jacobian variety in this direction:
{\it an indecomposable ppav $X$ is the Jacobian of a curve if and only if its
Kummer variety $K(X)$ has a trisecant line} and the solution of the
characterization problem of principally polarized Prym varieties. The latter
problem is almost as old and famous as the Riemann-Schottky problem but is much
harder. In some sense the Prym varieties may be geometrically the
easiest-to-understand ppavs beyond Jacobians, and studying them may be a first
step towards understanding the geometry of more general abelian varieties as
well.
  Our second and primary objective is to take this opportunity to elaborate on
motivations underlining the proposed solution of the Riemann-Schottky problem,
to introduce a certain circle of ideas and methods, developed in the theory of
soliton equations, and to convince the reader that they are algebro-geometric
in nature, simple and universal enough to be included in the Handbook of
moduli.
