  We consider the question of when the operation of contact surgery with
positive surgery coefficient, along a knot $K$ in a contact 3-manifold $Y$,
gives rise to a weakly fillable contact structure. We show that this happens if
and only if $Y$ itself is weakly fillable, and $K$ is isotopic to the boundary
of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if
$Y'$ is a contact manifold arising from positive contact surgery along $K$,
then any filling of $Y'$ is symplectomorphic to the complement of a suitable
neighborhood of such a disk in a filling of $Y$.
  Using this result we deduce several necessary conditions for a knot in the
standard 3-sphere to admit a fillable positive surgery, such as quasipositivity
and equality between the slice genus and the 4-dimensional clasp number, and we
give a characterization of such knots in terms of a quasipositive braid
expression. We show that knots arising as the closure of a positive braid
always admit a fillable positive surgery, as do knots that have lens space
surgeries, and suitable satellites of such knots. In fact the majority of
quasipositive knots with up to 10 crossings admit a fillable positive surgery.
On the other hand, in general (strong) quasipositivity, positivity, or
Lagrangian fillability need not imply a knot admits a fillable positive contact
surgery.
