We present an overview of the developments in the modeling of cardiac fluid/tissue mechanics and electromechanics in the framework of the project MATHCARD (http://www.mathcard.eu). In particular, we discuss the development of robust finite element methods and accurate mathematical models for the simulation of the cardiac excitation-contraction mechanisms at the cell and muscle levels. An integrated multiscale model encompasses a 4-field description of cardiac activity: electrophysiology, mechanical activation, large tissue deformations, and fluid dynamics. The link between active fiber contraction and the biochemical reactions at microscales is described by an active strain decomposition model. The implications of different modeling and algorithmic choices in the electromechanical behavior of cells and muscle are addressed in detail. We report on several preliminary results with a prototype fully integrated heart model aimed to study the main aspects of the overall cardiac function, including the interaction of electrophysiology, active biomechanics, as well as ventricular and arterial haemodynamics.