The purpose of this thesis is to study Bridgeland stability conditions on the bounded derived category of coherent sheaves Db (X) on a smooth projective complex variety X. The thesis is organized as follows. In Chapter 1, we outline the necessary background and preliminaries required for subsequent chapters. We begin with a review of categories and describe the construction of derived categories and triangulated categories. We also introduce the tilting technique which is important in constructing stability conditions in Chapter 3. Some basic definitions and theorems of sheaves and algebraic surfaces are present in Chapter 2. Some classical stabilities on algebraic surfaces are reviewed. In particular, Mumford-Takemoto stability will be used in Chapter2 as a part of defining stability conditions in Chapter 3.Chapter 3 is devoted to the stability conditions on categories - abelian categories and triangulated categories. We state that stability conditions on abelian categories are generalization of classical stabilities on varieties and give an overview of Bridgeland stability conditions following his paper [Bri07]. The aim of the last chapter is to study stability conditions on algebraic surfaces X. Under certain conditions, the Theorem 3.7.2 states that the stability function Z(ω,β) on tilted heart A ♯ (ω,β) has HN-property, and hence defines anumerical stability on Db (X).