This paper deals with the mathematical model that describes the function of the human heart. More specifically, it addresses the equations that express the electromechanical process, that is the mechanical deformation (contraction and relaxation) of the heart muscle that is induced by the electrical field that, at every heartbeat, is generated in the sino-atrial node and then propagates all across the cardiac cells. After deriving the equations of the mathematical model from basic physical principles, we proceed to their numerical approximations and discuss issues such as stability, accuracy and computational complexity. We close the paper by illustrating a few numerical results on test problems of potential interest for clinical applications.