  We study higher-order analogues of Dirac structures, extending the
multisymplectic structures that arise in field theory. We define higher Dirac
structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak
version of the usual lagrangian condition (which agrees with it only when
$k=1$). Higher Dirac structures transversal to $TM$ recover the higher Poisson
structures introduced in [8] as the infinitesimal counterparts of
multisymplectic groupoids. We describe the leafwise geometry underlying an
involutive isotropic subbundle in terms of a distinguished 1-cocycle in a
natural differential complex, generalizing the presymplectic foliation of a
Dirac structure. We also identify the global objects integrating higher Dirac
structures.
