The article provides a comprehensive review of noncommutative field theory, covering its fundamental concepts, mathematical foundations, and various applications. Noncommutative field theory is a generalization of conventional field theory where the coordinates of space-time commute only up to a nonvanishing parameter $\theta$. This theory arises naturally in limits of M-theory and string theory and has been studied extensively due to its connections with quantum Hall states and other physical phenomena. The review begins with the basics of noncommutative kinematics, including the algebraic structure, derivatives, and integrals. It then discusses the deformation of flat space-time and the emergence of noncommutativity through the deformation of the algebra of functions. The authors explore the symmetries of noncommutative spaces, such as translations and rotations, and introduce the concept of noncommutative tori. The article delves into the construction of field theory actions and symmetries, showing how standard Lagrangians can be generalized to noncommutative theories while preserving classical symmetries and Noether's theorem. It also examines gauge theories, highlighting the unification of space-time translations and gauge transformations in noncommutative theories. The authors discuss the challenges in defining local observables and introduce the concept of Wilson loops and open Wilson loops to address these issues. The review further explores the stress-energy tensor in noncommutative theories, noting its role in generating gauge transformations and the difficulties in defining a local conserved momentum density. It also discusses the introduction of fundamental matter fields and their transformation laws under noncommutative gauge transformations. Finally, the article touches on the applications of noncommutative field theory to the quantum Hall effect and its connections to string theory, including the role of noncommutativity in M-theory and string theory. The authors emphasize the importance of noncommutative field theory as a new universality class of theories and its potential for providing insights into quantum gravity and condensed matter physics.The article provides a comprehensive review of noncommutative field theory, covering its fundamental concepts, mathematical foundations, and various applications. Noncommutative field theory is a generalization of conventional field theory where the coordinates of space-time commute only up to a nonvanishing parameter $\theta$. This theory arises naturally in limits of M-theory and string theory and has been studied extensively due to its connections with quantum Hall states and other physical phenomena. The review begins with the basics of noncommutative kinematics, including the algebraic structure, derivatives, and integrals. It then discusses the deformation of flat space-time and the emergence of noncommutativity through the deformation of the algebra of functions. The authors explore the symmetries of noncommutative spaces, such as translations and rotations, and introduce the concept of noncommutative tori. The article delves into the construction of field theory actions and symmetries, showing how standard Lagrangians can be generalized to noncommutative theories while preserving classical symmetries and Noether's theorem. It also examines gauge theories, highlighting the unification of space-time translations and gauge transformations in noncommutative theories. The authors discuss the challenges in defining local observables and introduce the concept of Wilson loops and open Wilson loops to address these issues. The review further explores the stress-energy tensor in noncommutative theories, noting its role in generating gauge transformations and the difficulties in defining a local conserved momentum density. It also discusses the introduction of fundamental matter fields and their transformation laws under noncommutative gauge transformations. Finally, the article touches on the applications of noncommutative field theory to the quantum Hall effect and its connections to string theory, including the role of noncommutativity in M-theory and string theory. The authors emphasize the importance of noncommutative field theory as a new universality class of theories and its potential for providing insights into quantum gravity and condensed matter physics.