The heart occupies the most central role of the cardiovascular system, with the crucial role of supplying a continuous flow of blood through the vast network of vessels composing the circulatory system. Mathematical models of the heart in health and disease is an increasingly important tool for improving our understanding of this vital organ, and thereby help to reduce health problems and costs related to cardiovascular disorders. In this talk we give an overview of computational challenges and techniques related to modelling the mechanical function of the heart muscle. Models applied for this purpose vary in complexity from the simplest pressure-volume relations based on a given time varying elastance, to systems of differential equations that give a detailed description of cardiac electro-mechanical interaction. The primary focus of the talk will be on finite element modeling of the heart muscle, which includes modeling the passive mechanical properties of the tissue as well as the active muscle contraction and its coupling to electrophysiology. The heart muscle is normally modeled as a hyper-elastic material, with strongly non-linear and anisotropic material behavior. The resulting mathematical model consists of a large-strain elasticity equation, which is coupled to DAE systems that describe electro-chemical reactions on cell level, and also to a system of PDEs that describe the conduction of the electrical signal in the tissue. We discuss some approaches for solving this system, which offer a varying degree of coupling and feedback between electrophysiology and mechanics. We also discuss different approaches for coupling the heart muscle models to the rest of the circulatory system. The latter topic will also be covered in more detail in the other talks of the minisymposium.