Breast magnetic resonance imaging (MRI) and x-ray mammography are two image modalities widely used for the early detection and diagnosis of breast diseases in women. The combination of these modalities leads to a more accurate diagnosis and treatment of breast diseases. The aim of this paper is to review the registration between breast MRI and x-ray mammographic images using patient-specific finite element-based biomechanical models. Specifically, a biomechanical model is obtained from the patient's MRI volume and is subsequently used to mimic the mammographic acquisition. Due to the different patient positioning and movement restrictions applied in each image modality, the finite element analysis provides a realistic physics-based approach to perform the breast deformation. In contrast with other reviews, we do not only expose the overall process of compression and registration but we also include main ideas, describe challenges, and provide an overview of the used software in each step of the process. Extracting an accurate description from the MR images and preserving the stability during the finite element analysis require an accurate knowledge about the algorithms used, as well as the software and underlying physics. The wide perspective offered makes the paper suitable not only for expert researchers but also for graduate students and clinicians. We also include several medical applications in the paper, with the aim to fill the gap between the engineering and clinical performance.