This paper introduces the concept of a stability condition on a triangulated category, motivated by string theory and the study of Dirichlet branes. A stability condition consists of a central charge, a group homomorphism from the Grothendieck group to the complex numbers, and a collection of full additive subcategories indexed by real numbers, satisfying certain axioms. These conditions allow for the construction of a natural topology on the space of stability conditions, making it a manifold. The paper proves that this space is a manifold, possibly infinite-dimensional, and provides examples, including the case of the derived category of coherent sheaves on an elliptic curve, where the space of stability conditions is shown to be connected and homeomorphic to the universal cover of the group of positive definite 2x2 real matrices. The paper also discusses the relationship between stability conditions and t-structures, and introduces the notion of a slicing, which allows for finer decompositions of objects in a triangulated category. The paper concludes with the definition of a stability function and the Harder-Narasimhan filtration, which are essential tools in the study of stability conditions. The paper also introduces the concept of a locally-finite stability condition, which is crucial for proving the main theorem about the topology of the space of stability conditions.This paper introduces the concept of a stability condition on a triangulated category, motivated by string theory and the study of Dirichlet branes. A stability condition consists of a central charge, a group homomorphism from the Grothendieck group to the complex numbers, and a collection of full additive subcategories indexed by real numbers, satisfying certain axioms. These conditions allow for the construction of a natural topology on the space of stability conditions, making it a manifold. The paper proves that this space is a manifold, possibly infinite-dimensional, and provides examples, including the case of the derived category of coherent sheaves on an elliptic curve, where the space of stability conditions is shown to be connected and homeomorphic to the universal cover of the group of positive definite 2x2 real matrices. The paper also discusses the relationship between stability conditions and t-structures, and introduces the notion of a slicing, which allows for finer decompositions of objects in a triangulated category. The paper concludes with the definition of a stability function and the Harder-Narasimhan filtration, which are essential tools in the study of stability conditions. The paper also introduces the concept of a locally-finite stability condition, which is crucial for proving the main theorem about the topology of the space of stability conditions.