Shape Memory Alloys (SMAs) are nowadays widely exploited for the realization of innovative devices and have a great impact on the development of a variety of biomedical applications ranging from orthodontic archwires to vascular stents. The design, realization, and optimization of such devices are quite demanding tasks. Mathematics is involved in this process as a major tool in order to let the modeling more accurate, the numerical simulations more reliable, and the design more effective. Many material properties of SMAs such as martensitic reorientation, training, and ferromagnetic behavior, are still to be properly and efficiently addressed. Therefore, new modeling ideas, along with original analytical and numerical techniques, are required. This project is aimed at addressing novel mathematical issues in order to move from experimental materials results toward the solution of real-scale biomechanical Engineering problems. The research focus will be multidisciplinary and include modeling, analytic, numerical, and computational issues. A progress in the macroscopic description of SMAs, the computational simulation of real-scale SMA devices, and the optimization of the production processes will contribute to advance in the direction of innovative applications.