  We give a bordism-theoretic characterisation of those closed almost contact
(2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure.
Our method is to apply Eliashberg's h-principle for Stein manifolds in the
setting of Kreck's modified surgery. As an application, we show that any simply
connected almost contact 7-manifold with torsion free second homotopy group is
Stein fillable. We also discuss the Stein fillability of exotic spheres and
examine subcritical Stein fillability.
