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Heart is a vital organ that pumps blood to whole body. Synchronous contraction of the heart muscles assures that the required blood flow is supplied to organs. But sometimes the synchrony between those muscles is distorted, which results in reduced cardiac output that might lead to severe diseases, and even death. The most common of heart diseases are myocardial infarction and arrhythmias. The contraction of heart muscles is controlled by the electrical activity of the heart, therefore determination of that electrical activity could give us the information regarding the severeness and type of the disease. In order to diagnose heart diseases, classical 12 lead electrocardiogram (ECG) is the standard clinical tool. Although many cardiac diseases could be diagnosed with the 12 lead ECG, measurements from sparse electrode locations limit the interpretations. The main objective of this thesis is to determine the cardiac electrical activity from dense body surface measurements. This problem is called the inverse problem of electrocardiography. The high resolution maps of epicardial potentials could supply the physician the information that could not be obtained with any other method. But the calculation of those epicardial potentials are not easy / the problem is severely ill-posed due to the discretization and attenuation within the thorax. To overcome this ill-posedness, the solution should be constrained using prior information on the epicardial potential distributions. In this thesis, spatial and spatio-temporal Bayesian maximum a posteriori estimation (MAP), Tikhonov regularization and Kalman filter and Kalman smoother approaches are used to overcome the ill-posedness that is associated with the inverse problem of ECG. As part of the Kalman filter approach, the state transition matrix (STM) that determines the evolution of epicardial potentials over time is also estimated, both from the true epicardial potentials and previous estimates of the epicardial potentials. An activation time based approach was developed to overcome the computational complexity of the STM estimation problem. Another objective of this thesis is to study the effects of geometric errors to the solutions, and modify the inverse solution algorithms to minimize these effects. Geometric errors are simulated by changing the size and the location of the heart in the mathematical torso model. These errors are modeled as additive Gaussian noise in the inverse problem formulation. Residual-based and expectation maximization methods are implemented to estimate the measurement and process noise variances, as well as the geometric noise.