  We consider a large family F of torus bundles over the circle, and we use
recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact
structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first
Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein
fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic
or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and
fall into finitely many diffeomorphism classes. Moreover, for infinitely many
hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein
fillings of (Y,C).
