Submitted to arXiv on 3 Jun 2005 (v1), last revised 25 Jan 2006 (this version, v2).A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations, the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.J.V and F.C wish to thank the Research Foundation-Flanders (FWO) and the "Bijzonder Onderzoeksfonds" (BOF) of Ghent University, for financial support. They also thank the CSIC (Madrid) for its kind hospitality on the occasion of several research visits. M.dL and D.MdD wish to thank to MICYT (Spain) Grant MTM2004-7832 for financial support.Peer reviewed