This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from . Douglas’s work on Π-stability. From a mathematical point of view, the most interesting feature of the deﬁnition is that the set of stability conditions Stab(D) on a ﬁxed category D has a natural topology, thus deﬁning a new invariant of triangulated categories. In a separate article  I give a detailed description of this space of stability conditions in the case that D is the bounded derived category of coherent.