###### Abstract  
Brain source localization has been consistently implemented over the recent years to elucidate complex brain operations, pairing the high temporal resolution of the EEG with the high spatial estimation of the estimated sources. This review paper aims to present the basic principles of Electrical source imaging (ESI) in the context of the recent progress for solving the forward and the inverse problems, and highlight the advantages and limitations of the different approaches. As such, a synthesis of the current state-of-the-art methodological aspects is provided, offering a complete overview of the present advances with regard to the ESI solutions. Moreover, the new dimensions for the analysis of the brain processes are indicated in terms of clinical and cognitive ESI applications, while the prevailing challenges and limitations are thoroughly discussed, providing insights for future approaches that could help to alleviate methodological and technical shortcomings.  
EEG; source imaging; source localization; applications; challenges +
Footnote †: journal: _Signals_  
0000-0001-5000-0000-0000-00001-5000-0000-0000-0000-0000-0000-00000-000  
each individual (although to a small degree). In this context, the anatomical information as well as the corresponding electrode locations comprise the Leadfield matrix, the accurate calculation of which is closely related to the accuracy of the solutions of the inverse problem [6]. The EEG source localization has puzzled researchers since a vast variety of combinations of sources can produce the same signals recorded by the EEG [7]. This is a fundamental problem of EEG source localization, because even with a flawless head model and an excellent noiseless signal, the number of equations that can be solved is minuscule compared to the number of dipoles (source points) within the brain, leading to non-unique possible solutions. This non-uniqueness can be handled with mathematical, anatomical or neurophysiological constraints about the location and the direction of dipoles [8,9]. Therefore, even if non-uniqueness is not guaranteed, these a priori assumptions are utilized, in order to reduce the computational needs for the source estimation and increase the accuracy of the solutions.  
However, owing to recent advances which incorporate novel methodologies as well as the introduction of machine learning approaches in solving the inverse problem [10,11], the required time and the computational resources for the solution have been significantly reduced. Furthermore, sophisticated algorithms have diminished the localization error efficiently, estimating the location and activation of the different cortical regions [12]. This is evident in several applications, ranging from detecting and assisting the therapy of epilepsy, to interpreting emotions from facial expressions [6,13].  
This continually updated progress in both the forward and the inverse problem solutions, designates ESI as a \"living\" tool, revolutionizing neurodynamic analysis in clinical, biomedical and cognitive applications. On this premise, the presentation of the current advancements in EEG source localization have been the key motivation in this review paper, which aims to highlight the different approaches of the multiple state-of-the-art ESI algorithms, pinpointing the advantages and disadvantages of each method, as well as their related limitations that affect the spatial resolution or the computational complexity. Moreover, the recent clinical and cognitive applications of EEG source localization are discussed, as well as the implications and challenges occurring due to methodological and technical constraints, proposing future actions.