  We consider a compact manifold $(M,\mathfrak{F})$ with a foliation
$\mathfrak{F}$, and a smooth affine connection $\nabla$ on the tangent bundle
of the foliation $T\mathfrak{F}$. We introduce and study a foliated
completeness problem. Namely, under which conditions on $\nabla$ the leaves are
complete? We consider different natural geometric settings: the first one is
the case of a totally geodesic lightlike foliation of a compact Lorentzian
manifold, and the second one is the case where the leaves have particular
affine structures. In the first case, we characterize the completeness, and
obtain in particular that if a compact Lorentzian manifold admits a null
Killing field $V$ such that the distribution orthogonal to $V$ is integrable,
then it defines a (totally geodesic) foliation with complete leaves. In the
second case, we give a completeness result for a specific affine structure
called "the unimodular affine lightlike geometry", and characterize the
completeness for a natural relaxation of the geometry. On the other hand, we
study the global completeness of a compact Lorentzian manifold in the presence
of a null Killing field. We give two non-complete examples, starting from
dimension $3$: one is a locally homogeneous manifold, and the other is a $3$D
example where the Killing field dynamics is equicontinuous.
