We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot
K
K
in a contact 3-manifold
Y
Y
, gives rise to a weakly fillable contact structure. We show that this happens if and only if
Y
Y
itself is weakly fillable, and
K
K
is isotopic to the boundary of a properly embedded symplectic disk inside a filling of
Y
Y
. Moreover, if
Y
′
Y’
is a contact manifold arising from positive contact surgery along
K
K
, then any filling of
Y
′
Y’
is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of
Y
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.