We review some results about the theory of integrable dispersionless PDEs
arising as commutation condition of pairs of one-parameter families of vector
fields, developed by the authors during the last years. We review, in
particular, the formal aspects of a novel Inverse Spectral Transform including,
as inverse problem, a nonlinear Riemann - Hilbert (NRH) problem, allowing one
i) to solve the Cauchy problem for the target PDE; ii) to construct classes of
RH spectral data for which the NRH problem is exactly solvable; iii) to
construct the longtime behavior of the solutions of such PDE; iv) to establish
if a localized initial datum breaks at finite time. We also comment on the
existence of recursion operators and Backl\"und - Darboux transformations for
integrable dispersionless PDEs.Comment: 17 pages, 1 figure. Written rendition of the talk presented by one of
the authors (PMS) at the PMNP 2013 Conference, in a special session dedicated
to the memory of S. V. Manakov. To appear in the Proceedings of the
Conference PMNP 2013, IOP Conference Serie
