Taut foliations

We describe notions of tautness that arise in the study of C-0 foliations, C-1,C-0 or smoother foliations, and in geometry. We give examples to show that these notions are different. We prove that these variations of tautness are equivalent up to topological conjugacy, but their differences impact some classical foliation results. In particular, we construct examples of smoothly taut C-infinity,C-0 foliations that can be C-0 approximated by both weakly symplectically fillable, universally tight contact structures and by overtwisted contact structures.