This paper introduces the concept of a stability condition on a triangulated category, motivated by the study of Dirichlet branes in string theory and Douglas's work on II-stability. The key feature of this definition is that the set of stability conditions $\text{Stab}(\mathcal{D})$ on a fixed category $\mathcal{D}$ has a natural topology, making it a new invariant of triangulated categories. The paper proves that $\text{Stab}(\mathcal{D})$ is a manifold, possibly infinite-dimensional, and provides a detailed description of this space when $\mathcal{D}$ is the bounded derived category of coherent sheaves on a K3 surface. The introduction includes an example of a stability condition on a triangulated category, using the bounded derived category of coherent sheaves on a nonsingular projective curve $X$. The Harder-Narasimhan filtration and the phase function are defined, leading to the definition of a stability condition $(Z, \mathcal{P})$ on a triangulated category $\mathcal{D}$, consisting of a central charge $Z: K(\mathcal{D}) \to \mathbb{C}$ and a slicing $\mathcal{P}$ of $\mathcal{D}$. The paper also discusses the relationship between t-structures and stability conditions, showing that a stability condition can be constructed from a bounded t-structure and a stability function on its heart. Examples of stability conditions are provided, including one on the derived category of coherent sheaves on an elliptic curve and another on the derived category of finite-dimensional left modules over a finite-dimensional algebra.This paper introduces the concept of a stability condition on a triangulated category, motivated by the study of Dirichlet branes in string theory and Douglas's work on II-stability. The key feature of this definition is that the set of stability conditions $\text{Stab}(\mathcal{D})$ on a fixed category $\mathcal{D}$ has a natural topology, making it a new invariant of triangulated categories. The paper proves that $\text{Stab}(\mathcal{D})$ is a manifold, possibly infinite-dimensional, and provides a detailed description of this space when $\mathcal{D}$ is the bounded derived category of coherent sheaves on a K3 surface. The introduction includes an example of a stability condition on a triangulated category, using the bounded derived category of coherent sheaves on a nonsingular projective curve $X$. The Harder-Narasimhan filtration and the phase function are defined, leading to the definition of a stability condition $(Z, \mathcal{P})$ on a triangulated category $\mathcal{D}$, consisting of a central charge $Z: K(\mathcal{D}) \to \mathbb{C}$ and a slicing $\mathcal{P}$ of $\mathcal{D}$. The paper also discusses the relationship between t-structures and stability conditions, showing that a stability condition can be constructed from a bounded t-structure and a stability function on its heart. Examples of stability conditions are provided, including one on the derived category of coherent sheaves on an elliptic curve and another on the derived category of finite-dimensional left modules over a finite-dimensional algebra.