Mental disorders represent critical public health challenges as they are leading contributors to the global burden of disease and intensely influence social and financial welfare of individuals. The present comprehensive review concentrate on the two mental disorders: Major depressive Disorder (MDD) and Bipolar Disorder (BD) with noteworthy publications during the last ten years. There is a big need nowadays for phenotypic characterization of psychiatric disorders with biomarkers. Electroencephalography (EEG) signals could offer a rich signature for MDD and BD and then they could improve understanding of pathophysiological mechanisms underling these mental disorders. In this review, we focus on the literature works adopting neural networks fed by EEG signals. Among those studies using EEG and neural networks, we have discussed a variety of EEG based protocols, biomarkers and public datasets for depression and bipolar disorder detection. We conclude with a discussion and valuable recommendations that will help to improve the reliability of developed models and for more accurate and more deterministic computational intelligence based systems in psychiatry. This review will prove to be a structured and valuable initial point for the researchers working on depression and bipolar disorders recognition by using EEG signals.