Thesis advisor: John A. Baldwin / We study the relationship between knots in contact three-manifolds and complex curves in Stein surfaces. To do so, we extend the notion of quasipositivity from classical braids to links that are braided with respect to an open book decomposition of an arbitrary closed, oriented three-manifold. Our main results characterize the transverse links in Stein-fillable contact three-manifolds that bound smooth holomorphic curves in Stein fillings. This characterization is made possible by new techniques in the theory of characteristic and open book foliations on surfaces in three-manifolds. We also explore the Seifert genera of cross-sections of complex plane curves, minimal braid representatives of quasipositive links, and the relationship between Legendrian ribbons in contact three-manifolds and strongly quasipositive braids with respect to compatible open books. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.