In recent years an extensive use of social networking platforms has been registered, coupled with the increasing popularity of wearable devices, which is expected to double within the next four years (Source: Gartner August 2017). Physical tracking activities and the publishing of peoples own images, emoticons, audio files and texts on social platforms have become daily practices, with an increase in the availability of data, and therefore potential information, for each user. To extract knowledge from this data, new computational technologies such as Sentiment Analysis (SA) and Affective Computing (AC) have found applications in fields such as marketing, politics, social sciences, cognitive sciences, medical sciences, etc. Such technologies aim to automatically extract emotions from heterogeneous data sources such as text, images, audio, video, and a plethora of biosignals such as voice, facial expression, electroencephalographic signals (EEG), gestures, etc. and find application in various fields. The paper introduces main concepts of Sentiment Analysis and Affective Computing and presents an overview of the primary methodologies and techniques used to recognize emotions from the analysis of various data sources such as text, images, voice signals, EEG. Finally, the paper discusses various applications of those techniques to neurosciences and underlines the high-performance issues of SA and AC, as well as challenges and future trends.