Let (X, J) be a closed, connected almost complex four-manifold. Let

be the complement of an open disc in X and let

be the contact structure on the boundary

which is compatible with a symplectic structure on

, Then we show that (X, J) is symplectic if and only if the contact structure

on

is isomorphic to the standard contact structure on the 3-sphere

and

is J-concave. Also we show that there is a contact structure

which is not strongly symplectically fillable but symplectically fillable, and that

has infinitely many non-diffeomorphic minimal fillings whose restrictions on

are

where

is the restriction of the standard symplectic structure on

.