Internal organ motion during radiation delivery may lead to underdosing of cancer cells or overdosing of normal tissue, potentially causing treatment failure or normal-tissue toxicity. Organ motion is of particular concern in the treatment of lung and abdominal cancers, where breathing induces large tumor displacement and organ deformation. A new generation of radiotherapy devices is equipped with on-board MRI scanners to acquire a real-time movie of the patient's anatomy during radiation delivery. The goal of this research is to develop, calibrate, and test motion predictive models that employ real-time MRI images to provide the short-term trajectory of respiration-induced anatomical motion during radiation delivery. A semi-Markov model predicts transitions between the phases of a respiratory cycle, and a Markov model predicts transitions to future respiratory cycles, leading to accurate motion forecasting over longer-term horizons. The intended application for this work is real-time tracking and re-optimization of intensity-modulated radiation delivery.