  We consider slice disks for knots in the boundary of a smooth compact
4-manifold $X^{4}$. We call a knot $K \subset \partial X$ deep slice in $X$ if
there is a smooth properly embedded 2-disk in $X$ with boundary $K$, but $K$ is
not concordant to the unknot in a collar neighborhood $\partial X \times I$ of
the boundary. We point out how this concept relates to various well-known
conjectures and give some criteria for the nonexistence of such deep slice
knots. Then we show, using the Wall self-intersection invariant and a result of
Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of
2-handles always has a deep slice knot in the boundary. We end by considering
4-manifolds where every knot in the boundary bounds an embedded disk in the
interior. A generalization of the Murasugi-Tristram inequality is used to show
that there does not exist a compact, oriented 4-manifold $V$ with spherical
boundary such that every knot $K \subset S^3 = \partial V$ is slice in $V$ via
a null-homologous disk.
