Abstract: The digital twin concept has its origins in industry. One industrial equipment manufacturer advertised its digital twin capabilities to its customers as ”No unplanned downtime” for its products. There is a compelling aspirational analog in healthcare: ``No unplanned doctor visits." Of course, the challenges of building digital twins for human patients are incomparably greater than for machinery. Nonetheless, there are now several instances of what might be called digital twins in medicine, and many more ongoing development projects. Aside from our incomplete understanding of human biology, relative sparseness of data characterizing human patients, and logistical difficulties in implementing computational models in healthcare, there are many mathematical and computational problems that need to be solved. Examples include calibration and validation of multiscale, hybrid, stochastic computational models, forecasting algorithms, and optimal control methods. This talk will describe some of these problems and outline a mathematical research program for the field.