Digital Twins (DTs) are one of the disruptive technologies associated with the Industry 4.0 concept. A DT connects the physical manufacturing system with the digital cyberspace, via the synchronization of the simulation (i.e., physical configurations) and data models (i.e., product, process, and resource models) of the manufacturing system. This synchronization of both worlds—the physical and digital—allows one to address the issue of manufacturing customized products. This challenge of mass customization (1) puts forward the goal of achieving the highest level of customer satisfaction, and (2) creates the need for the optimization of the complete value creation process. Within an Industry 4.0 context, the latter is translated as the interlinking of production resources and systems, via a DT, as it is in the physical world where the actual value-creation process takes place. The success of an Industry 4.0 mass customization environment (or mass customization 4.0), depends on its degree/level of sustainability. For these reasons, the present paper presents a review of relevant concepts related to the role of DTs in the achievement of a mass customization 4.0 environment, plus some proposals of how to address the identified research challenges. A future research agenda is proposed at the end of the paper.