The paper explores the role of the gravitational sector in the Standard-Model Extension (SME), which incorporates Lorentz and CPT violations. It develops a framework for addressing this in Riemann-Cartan spacetimes, which include Riemann and Minkowski geometries as limiting cases. The SME is studied in a Riemann-Cartan background, with leading-order terms in the action involving operators of mass dimensions three and four constructed. The paper discusses how arbitrary Lorentz and CPT violations can be incorporated into general relativity and other gravity theories based on Riemann-Cartan geometries. It provides the dominant terms in the effective low-energy action for the gravitational sector, completing the formulation of the leading-order SME with gravity. Explicit Lorentz symmetry breaking is incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this difficulty. The SME is a framework that incorporates both general relativity and the Standard Model (SM) of particle physics, with terms describing arbitrary coordinate-independent Lorentz violations. In the Minkowski-spacetime limit, the dominant terms in the SME action include the pure-gravity and minimally coupled SM actions, along with leading-order terms introducing Lorentz symmetry violations. The SME has been extensively studied in the Minkowski-spacetime limit, where all terms expected to dominate at low energies are known. The paper aims to construct explicitly the modifications appearing in non-Minkowski spacetimes, including those in the pure-gravity sector and those involving gravitational couplings in the matter and gauge sectors. The paper discusses the framework for local Lorentz violations, including the vierbein formalism, which incorporates spinors and distinguishes between local Lorentz and general coordinate transformations. The presence of Lorentz violation in a local Lorentz frame is signaled by a nonzero vacuum value for one or more quantities carrying local Lorentz indices, called coefficients for Lorentz violation. The paper also discusses the action and covariant conservation laws in the presence of Lorentz violation, showing how the energy-momentum tensor and symmetry properties are modified. The paper presents the explicit form of the two partial actions for the QED extension and some of their basic physical implications. It discusses the fermion and photon sectors, showing how the SME incorporates Lorentz and CPT violations in the presence of gravitational couplings. The paper concludes with a summary of the key findings and implications of the study.The paper explores the role of the gravitational sector in the Standard-Model Extension (SME), which incorporates Lorentz and CPT violations. It develops a framework for addressing this in Riemann-Cartan spacetimes, which include Riemann and Minkowski geometries as limiting cases. The SME is studied in a Riemann-Cartan background, with leading-order terms in the action involving operators of mass dimensions three and four constructed. The paper discusses how arbitrary Lorentz and CPT violations can be incorporated into general relativity and other gravity theories based on Riemann-Cartan geometries. It provides the dominant terms in the effective low-energy action for the gravitational sector, completing the formulation of the leading-order SME with gravity. Explicit Lorentz symmetry breaking is incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this difficulty. The SME is a framework that incorporates both general relativity and the Standard Model (SM) of particle physics, with terms describing arbitrary coordinate-independent Lorentz violations. In the Minkowski-spacetime limit, the dominant terms in the SME action include the pure-gravity and minimally coupled SM actions, along with leading-order terms introducing Lorentz symmetry violations. The SME has been extensively studied in the Minkowski-spacetime limit, where all terms expected to dominate at low energies are known. The paper aims to construct explicitly the modifications appearing in non-Minkowski spacetimes, including those in the pure-gravity sector and those involving gravitational couplings in the matter and gauge sectors. The paper discusses the framework for local Lorentz violations, including the vierbein formalism, which incorporates spinors and distinguishes between local Lorentz and general coordinate transformations. The presence of Lorentz violation in a local Lorentz frame is signaled by a nonzero vacuum value for one or more quantities carrying local Lorentz indices, called coefficients for Lorentz violation. The paper also discusses the action and covariant conservation laws in the presence of Lorentz violation, showing how the energy-momentum tensor and symmetry properties are modified. The paper presents the explicit form of the two partial actions for the QED extension and some of their basic physical implications. It discusses the fermion and photon sectors, showing how the SME incorporates Lorentz and CPT violations in the presence of gravitational couplings. The paper concludes with a summary of the key findings and implications of the study.