  If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a
foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and
such that $A/B$ is locally equivalent with Lie algebroids over the slice
manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie
algebroids, we discuss the following subjects: the dual Poisson structure and
Vaintrob's super-vector field, cohomology and deformations of the foliation,
integration to a Lie groupoid. In the last section, we define a corresponding
notion of a foliation of a Courant algebroid $A$ as a bracket-closed, isotropic
subbundle $B$ with anchor image $T\mathcal{F}$ and such that $B^\perp/B$ is
locally equivalent with Courant algebroids over the slice manifolds of
$\mathcal{F}$. Examples that motivate the definition are given.
