The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.This work has been supported by grant BFM2001-2272 from the Ministry of Science and Technology. A. Santamaría–Merino wishes to thank the Programa de formación de Investigadores of the Departamento de Educación, Universidades e Investigación of the Basque Government (Spain) for financial support.Peer reviewed