In a demand context of mass customization, shifting towards the mass personalization of products, assembly operations face the trade-off between highly productive automated systems and flexible manual operators. Novel digital technologies—conceptualized as Industry 4.0—suggest the possibility of simultaneously achieving superior productivity and flexibility. This article aims to address how Industry 4.0 technologies could improve the productivity, flexibility and quality of assembly operations. A systematic literature review was carried out, including 234 peer-reviewed articles from 2010–2020. As a result, the analysis was structured addressing four sets of research questions regarding (1) assembly for mass customization; (2) Industry 4.0 and performance evaluation; (3) Lean production as a starting point for smart factories, and (4) the implications of Industry 4.0 for people in assembly operations. It was found that mass customization brings great complexity that needs to be addressed at different levels from a holistic point of view; that Industry 4.0 offers powerful tools to achieve superior productivity and flexibility in assembly; that Lean is a great starting point for implementing such changes; and that people need to be considered central to Assembly 4.0. Developing methodologies for implementing Industry 4.0 to achieve specific business goals remains an open research topic.