<p>The Digital Twin, a virtual representation of a physical object that mimics its structure and behavior to inform decisions and optimize operational efficiency, is an established paradigm in industry. While Modelling, Simulation, and Optimization have been a standard practice in industry since long, the complementary role of models and data as well as a holistic and life cycle spanning approach distinguishes the Digital Twin paradigm. However, albeit nearly every industry is highlighting the potential and strategic importance of Digital Twins, they are by far not at a level of industrial practice as publicity suggests. A major hurdle is the effort associated with creating actionable and impactful Digital Twins. The reasons are multi-fold but novel algorithms and mathematical concepts will be key to overcome many of these. Within this article, we review the concept of the Digital Twin, major challenges of fostering its adoption, and highlight how research in Applied and Industrial Mathematics is key to address corresponding roadblocks.</p>