Inspired by tau-tilting theory [3], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra Lambda, we give bijections between two-term tilting complexes in K-b(proj Lambda), nu-stable support tau-tilting Lambda-modules, and nu-stable functorially finite torsion classes in mod Lambda. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in l if Lambda is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category l. We also study some properties of support tau-tilting modules over 2-CY tilted algebras, and we give a necessary condition such that algebras are 2-CY tilted in terms of support tau-tilting modules.