The integration of magnetic resonance imaging (MRI) for guidance in external beam radiotherapy has faced significant research and development efforts in recent years. The current availability of linear accelerators with an embedded MRI unit, providing volumetric imaging at excellent soft tissue contrast, is expected to provide novel possibilities in the implementation of image-guided adaptive radiotherapy (IGART) protocols. This study reviews open medical physics issues in MR-guided radiotherapy (MRgRT) implementation, with a focus on current approaches and on the potential for innovation in IGART. Daily imaging in MRgRT provides the ability to visualize the static anatomy, to capture internal tumor motion and to extract quantitative image features for treatment verification and monitoring. Those capabilities enable the use of treatment adaptation, with potential benefits in terms of personalized medicine. The use of online MRI requires dedicated efforts to perform accurate dose measurements and calculations, due to the presence of magnetic fields. Likewise, MRgRT requires dedicated quality assurance (QA) protocols for safe clinical implementation. Reaction to anatomical changes in MRgRT, as visualized on daily images, demands for treatment adaptation concepts, with stringent requirements in terms of fast and accurate validation before the treatment fraction can be delivered. This entails specific challenges in terms of treatment workflow optimization, QA, and verification of the expected delivered dose while the patient is in treatment position. Those challenges require specialized medical physics developments towards the aim of fully exploiting MRI capabilities. Conversely, the use of MRgRT allows for higher confidence in tumor targeting and organs-at-risk (OAR) sparing. The systematic use of MRgRT brings the possibility of leveraging IGART methods for the optimization of tumor targeting and quantitative treatment verification. Although several challenges exist, the intrinsic benefits of MRgRT will provide a deeper understanding of dose delivery effects on an individual basis, with the potential for further treatment personalization.