Meaningful computer based simulations of the electrocardiogram (ECG), linking models of the electrical activity of the heart to ECG signals, are a necessary step towards the development of personalized cardiac models from clinical data. An ECG simulator is, in addition, a valuable tool for building a virtual data base of pathological conditions, to test and train medical devices but also to improve the knowledge on the clinical significance of some ECG signals. In the present work, we show that meaningful ECG simulations (in normal or pathological conditions) can be obtained with a coupled heart-torso mathematical model fully based on partial differential equations: a reaction diffusion system in the heart (called bidomain model) and the Laplace equation in the torso.These equations are coupled on the heart-torso interface to obtain the ECG model. We present different numerical schemes that could be used to solve this complex multi-scale problem, we also propose different strategies allowing to reduce the computational cost of the ECG simulator.These strategies are based on space uncoupling using domain-decomposition methods and state variables uncoupling based on Jacobi like and Gauss-Seidel like time splitting schemes. As an example of applications we use the ECG simulator to build a data base for solving the inverse problem in electrocardiography. This problem is a major concern for the LIRYC (L'Institut de RYthmologie et modélisation Cardiaque) since it allows to non-invasivaly construct the electrical potential on the heart surface based on measurement on the body surface. We present different approches based on a Poincaré?Steklov formulation of the inverse problem and show the numerical results.