This book contains the lectures given at the Centro Internazionale Matematico Estivo (CIME), during the session Direct and Inverse Method in Non Linear Evolution Equations, held at Cetraro in September 1999. The lecturers were R. Conte of the Service de physique de l’´etat condens´e, CEA Saclay, F. Magri of the University of Milan, M. Musette of Dienst Theoretical Naturalness, Verite Universities Brussels, J. Satsuma of the Graduate School of Mathematical Sciences, University of Tokyo and P. Winternitz of the Centre de recherches math´ematiques, Universit´e de Montr´eal
