  We present an introduction to the theory of dispersive shock waves in the
framework of the approach proposed by Gurevich and Pitaevskii (Zh. Eksp. Teor.
Fiz., 65, 590 (1973) [Sov. Phys. JETP, 38, 291 (1974)]) based on the Whitham
theory of modulation of nonlinear waves. We explain how Whitham equations for a
periodic solution can be derived for the Korteweg-de Vries equation and outline
some elementary methods to solve them. We illustrate this approach with
solutions to the main problems discussed by Gurevich and Pitaevskii. We
consider a generalization of the theory to systems with weak dissipation and
discuss the theory of dispersive shock waves for the Gross-Pitaevskii equation.
