The notions of partial actions and partial representations have been extensively studied in several algebraic contexts in the last 25 years. In this talk, we introduce these concepts and give a short overview of the results known for finite groups. We will briefly show how this theory extends naturally to the classical global theory, in particular in the important case of the symmetric group. This is joint work with William Hautekiet, Paolo Saracco, and Joost Vercruysse.