In this dissertation I propose a new source imaging algorithm which uses surface non-invasive measurements such as EEG and MEG to estimate underlying brain activities. I employ sparse signal processing techniques (imposing sparsity in multiple domains) as well as an iterative reweighting scheme to come up with an algorithm which objectively and without the need of applying any subjective thresholds, yields extended solutions that not only precisely estimates the location of underlying sources of activity in the brain, but also provides a high quality estimate of the underlying sources’ extent and size. This algorithm is formulated as a convex optimization problem. I have also proposed a scheme to further develop this algorithm to become suitable for imaging sources that evolve over time, basically providing a spatio-temporal estimate of underlying brain activity. In this manner an efficient algorithm that can image dynamic underlying brain networks is developed. The main application this algorithm was motivated by and validated in, is imaging the epileptogenic tissue in focal epilepsy patients. It is shown in this dissertation through analyzing inter-ictal spikes and ictal signals from the EEG recordings of focal epilepsy patients and subsequently comparing it to clinical findings in these patients that the proposed algorithm works well in real-world applications and clinical settings. These clinical findings included the surgically resected volume and seizure onset zone identified by intra-cranial studies; our estimated epileptogenic tissue matched these clinical findings well. While this algorithm was developed for and tested in this particular application, i.e. epileptic activity imaging, it is a general source imaging algorithm and many other applications are also possible, as will be pointed out throughout this dissertation.