This paper introduces the concept of the intrinsic normal cone for Deligne-Mumford stacks, which is a key tool in the study of moduli spaces in algebraic geometry. The intrinsic normal cone is an algebraic stack associated with a Deligne-Mumford stack X, and it captures information about the singularities of X. The construction of the intrinsic normal cone is based on the cotangent complex of X and involves the notion of cone stacks, which are Artin stacks that can be locally described as the quotient of a cone by a vector bundle action. The paper also develops the theory of obstruction theories for Deligne-Mumford stacks. An obstruction theory is a morphism of complexes in the derived category of sheaves on X that satisfies certain conditions, and it allows the definition of a virtual fundamental class of X. This virtual fundamental class is a class in the Chow ring of X that has the expected dimension, which is a lower bound for the dimension of X at each point. The intrinsic normal cone is used to construct the virtual fundamental class, and the paper provides a detailed construction of this cone and its properties. The intrinsic normal cone is shown to be a closed subcone stack of the normal sheaf of X, which is a stack associated with the normal bundle of X. The paper also discusses the relationship between the intrinsic normal cone and the deformations of affine X-schemes, showing that the intrinsic normal cone carries obstructions for such deformations. This motivates the introduction of obstruction theories for X, which are objects in the derived category of sheaves on X together with a morphism to the cotangent complex of X, satisfying certain conditions. The paper concludes with applications of the results to various moduli problems, including the construction of Gromov-Witten invariants and the study of moduli stacks of projective varieties. The intrinsic normal cone and obstruction theories provide a powerful framework for understanding the geometry of moduli spaces and their virtual fundamental classes.This paper introduces the concept of the intrinsic normal cone for Deligne-Mumford stacks, which is a key tool in the study of moduli spaces in algebraic geometry. The intrinsic normal cone is an algebraic stack associated with a Deligne-Mumford stack X, and it captures information about the singularities of X. The construction of the intrinsic normal cone is based on the cotangent complex of X and involves the notion of cone stacks, which are Artin stacks that can be locally described as the quotient of a cone by a vector bundle action. The paper also develops the theory of obstruction theories for Deligne-Mumford stacks. An obstruction theory is a morphism of complexes in the derived category of sheaves on X that satisfies certain conditions, and it allows the definition of a virtual fundamental class of X. This virtual fundamental class is a class in the Chow ring of X that has the expected dimension, which is a lower bound for the dimension of X at each point. The intrinsic normal cone is used to construct the virtual fundamental class, and the paper provides a detailed construction of this cone and its properties. The intrinsic normal cone is shown to be a closed subcone stack of the normal sheaf of X, which is a stack associated with the normal bundle of X. The paper also discusses the relationship between the intrinsic normal cone and the deformations of affine X-schemes, showing that the intrinsic normal cone carries obstructions for such deformations. This motivates the introduction of obstruction theories for X, which are objects in the derived category of sheaves on X together with a morphism to the cotangent complex of X, satisfying certain conditions. The paper concludes with applications of the results to various moduli problems, including the construction of Gromov-Witten invariants and the study of moduli stacks of projective varieties. The intrinsic normal cone and obstruction theories provide a powerful framework for understanding the geometry of moduli spaces and their virtual fundamental classes.