The ultimate potential of the MR-linac is real-time adaptive MR-guided radiotherapy (aMRgRT), i.e. adapt the radiation plan in real-time according to real-time 3D motion estimates. One of the major technical roadblocks towards achieving this goal is the real-time 3D motion estimation. This thesis presents two new approaches in this context. The main method is called MR-MOTUS and is the subject of Chapters 2-4. All methods in this thesis are motivated by the observation that internal body motion exhibits a high level of spatio-temporal correlation, and could be reconstructed from minimal MRI-data that can be acquired in real-time. Chapter 2 demonstrates the proof of concept. The MR-MOTUS signal model was derived that explicitly relates motion-fields and a reference image to k-space data, and the minimization problem was formulated to reconstruct these motion-fields from the data. The proof-of-concept was demonstrated by reconstructions of in vivo 3D rigid head motion and 3D non-rigid respiratory motion from retrospectively highly undersampled k-space data, and 2D non-rigid respiratory motion-field reconstruction on prospectively undersampled data. Chapter 3 introduces several improvements to tighten the gap towards clinical application, and extends the framework to 3D+t spatio-temporal motion-field reconstructions by introducing a low-rank motion model, which naturally separates motion-fields into two components: a spatial component, and a temporal component. This model reduced the number of unknowns for space-time motion-fields by two orders of magnitude, and thereby enabled 3D+t motion-field reconstruction with high temporal resolution on a desktop PC. However, just high temporal resolution is not sufficient; the reconstructions also need to be available in real-time during the treatments. In Chapter 4, the previous reconstructions were therefore extended to real-time reconstructions at 6.7 Hz using a two-step approach, which leverages the low-rank separation of motion-fields into spatial and temporal components. In the first phase, the spatial component is assumed to be fixed in time over several minutes and is reconstructed with an offline reconstruction. In the second phase, the temporal component that encodes the dynamics in the motion-field is reconstructed per dynamic in an online reconstruction. The main rationale behind this approach is that the temporal component has few degrees of freedom (<10), and can be reconstructed in real-time from rapidly acquired k-space data if the spatial component is available. The reconstructions above assume a similarity between motion in the training and inference phase, which need not be true in practice. In Chapter 5, we therefore propose a probabilistic framework for real-time motion estimation with a measure of estimation uncertainty, based on the machine learning theory of Gaussian Processes (GPs). A GP was trained to infer the most-likely motion-field representation coefficients from few k-space data, with a frame-rate of 69 3D motion-fields-per-second. Additionally, the estimation uncertainty was used to design a rejection criterion to flag dynamics with potentially erroneous motion estimates. This strategy preserved low end-point-errors (75th percentiles ≤ 0.88 mm) during simulations and detected abnormal motion in healthy volunteers. Such detection of potentially erroneous motion estimates could play a crucial role to ensure patient safety during real-time adaptive MRgRT.