The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly 2 cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exactly two complements. We generalize (support) tilting modules to what we call (support) tau-tilting modules, and show that an almost support tau-tilting module has exactly two complements for any finite dimensional algebra. For a finite dimensional k-algebra A, we establish bijections between functorially finite torsion classes in mod A, support tau-tilting modules and two-term silting complexes in Kb(proj A). Moreover these objects correspond bijectively to cluster-tilting objects in C if A is a 2-CY tilted algebra associated with a 2-CY triangulated category C. As an application, we show that the property of 2 complements holds also for two-term silting complexes in Kb(proj A).