We study the behavior of different versions of the Ozsvath-Szabo tau-invariant for holomorphically fillable links in Stein domains. More specifically, we relate the Hedden's version of the invariant, which needs the assumption that our links live in a contact 3-manifold with non-vanishing contact invariant, with the one introduced by Grigsby, Ruberman and Strle, which on the other hand only depends on the pair link-Spin^c 3-manifold and is then a purely topological invariant. This is joint work with Antonio Alfieri. The main goal of the talk is to describe how our work allows us to recover results about properly embedded pseudo-holomorphic curves, such as the slice-Bennequin inequality and the relative Thom conjecture, and to find new restrictions on the topology of Stein fillings of certain 3-manifolds.