This paper reviews the fundamental concepts of backpropagation, a widely used method for calculating derivatives in artificial neural networks and other systems. It begins by introducing basic backpropagation, which is commonly used in pattern recognition and fault diagnosis. The paper then presents the equations for backpropagation through time and discusses its applications in areas such as pattern recognition, dynamic systems, system identification, and control. It highlights the advantages of this method, including its ability to handle systems with simultaneous equations or true recurrent networks. The paper also addresses practical issues and provides pseudocode to clarify the algorithms. The core of backpropagation is an efficient and exact method for calculating derivatives of a target quantity with respect to input quantities, making it suitable for a wide range of applications beyond neural networks. The paper concludes by discussing extensions of the method, including its use in other types of networks and applications such as neuroidentification and neurocontrol.This paper reviews the fundamental concepts of backpropagation, a widely used method for calculating derivatives in artificial neural networks and other systems. It begins by introducing basic backpropagation, which is commonly used in pattern recognition and fault diagnosis. The paper then presents the equations for backpropagation through time and discusses its applications in areas such as pattern recognition, dynamic systems, system identification, and control. It highlights the advantages of this method, including its ability to handle systems with simultaneous equations or true recurrent networks. The paper also addresses practical issues and provides pseudocode to clarify the algorithms. The core of backpropagation is an efficient and exact method for calculating derivatives of a target quantity with respect to input quantities, making it suitable for a wide range of applications beyond neural networks. The paper concludes by discussing extensions of the method, including its use in other types of networks and applications such as neuroidentification and neurocontrol.