Patient motion is an important consideration in modern PET image reconstruction. Advances in PET technology mean motion has an increasingly important influence on resulting image quality. Motion-induced artifacts can have adverse effects on clinical outcomes, including missed diagnoses and oversized radiotherapy treatment volumes. This review aims to summarize the wide variety of motion correction techniques available in PET and combined PET/CT and PET/MR, with a focus on the latter. A general framework for the motion correction of PET images is presented, consisting of acquisition, modeling, and correction stages. Methods for measuring, modeling, and correcting motion and associated artifacts, both in literature and commercially available, are presented, and their relative merits are contrasted. Identified limitations of current methods include modeling of aperiodic and/or unpredictable motion, attaining adequate temporal resolution for motion correction in dynamic kinetic modeling acquisitions, and maintaining availability of the MR in PET/MR scans for diagnostic acquisitions. Finally, avenues for future investigation are discussed, with a focus on improvements that could improve PET image quality, and that are practical in the clinical environment.