This paper presents a systematic study of entanglement entropy in relativistic quantum field theory. The entanglement entropy $ S_A $ is defined as the von Neumann entropy $ S_A = -\operatorname{Tr} \rho_A \log \rho_A $, corresponding to the reduced density matrix $ \rho_A $ of a subsystem A. For a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge $ c $, the result $ S_A \sim (c/3) \log \ell $ is re-derived when A is a finite interval of length $ \ell $ in an infinite system. This result is extended to many other cases, including finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For a system away from its critical point, when the correlation length $ \xi $ is large but finite, it is shown that $ S_A \sim \mathcal{A}(c/6) \log \xi $, where A is the number of boundary points of A. These results are verified for a free massive field theory and used to confirm a scaling ansatz for the case of finite-size off-critical systems, as well as for integrable lattice models like the Ising and XXZ models. The free-field results are extended to higher dimensions, motivating a scaling form for the singular part of the entanglement entropy near a quantum phase transition. The paper also discusses the entanglement entropy in 2D conformal field theory, the entropy in non-critical 1+1-dimensional models, and the use of the corner transfer matrix method for integrable models. The analysis shows that the entanglement entropy depends on the central charge $ c $, the correlation length $ \xi $, and the geometry of the system. The results are consistent with the expected behavior of entanglement entropy near quantum phase transitions and provide a framework for understanding the scaling properties of entanglement entropy in various quantum systems.This paper presents a systematic study of entanglement entropy in relativistic quantum field theory. The entanglement entropy $ S_A $ is defined as the von Neumann entropy $ S_A = -\operatorname{Tr} \rho_A \log \rho_A $, corresponding to the reduced density matrix $ \rho_A $ of a subsystem A. For a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge $ c $, the result $ S_A \sim (c/3) \log \ell $ is re-derived when A is a finite interval of length $ \ell $ in an infinite system. This result is extended to many other cases, including finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For a system away from its critical point, when the correlation length $ \xi $ is large but finite, it is shown that $ S_A \sim \mathcal{A}(c/6) \log \xi $, where A is the number of boundary points of A. These results are verified for a free massive field theory and used to confirm a scaling ansatz for the case of finite-size off-critical systems, as well as for integrable lattice models like the Ising and XXZ models. The free-field results are extended to higher dimensions, motivating a scaling form for the singular part of the entanglement entropy near a quantum phase transition. The paper also discusses the entanglement entropy in 2D conformal field theory, the entropy in non-critical 1+1-dimensional models, and the use of the corner transfer matrix method for integrable models. The analysis shows that the entanglement entropy depends on the central charge $ c $, the correlation length $ \xi $, and the geometry of the system. The results are consistent with the expected behavior of entanglement entropy near quantum phase transitions and provide a framework for understanding the scaling properties of entanglement entropy in various quantum systems.