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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  ∈  F   , a Stein fillable contact structure    ξ  Y   . We prove that (i) each Stein filling of    (  Y  ,   ξ  Y   )    has vanishing first Chern class and first Betti number, (ii) if    Y  ∈  F    is elliptic, then all Stein fillings of    (  Y  ,   ξ  Y   )    are pairwise diffeomorphic and (iii) if    Y  ∈  F    is parabolic or hyperbolic, then all Stein fillings of    (  Y  ,   ξ  Y   )    share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles    Y  ∈  F    we exhibit non‐homotopy equivalent Stein fillings of    (  Y  ,   ξ  Y   )   .

On Stein fillings of contact torus bundles