  In this note, we classify Stein fillings of an infinite family of contact
3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can
be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain
Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic
deformation, of an infinite family of contact 3-manifolds which can be obtained
by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian twist
knots. As a corollary, we obtain a classification of Stein fillings of an
infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.
