Subject motion in MRI is a relevant problem in the daily clinical routine as well as in scientific studies. Since the beginning of clinical use of MRI, many research groups have developed methods to suppress or correct motion artefacts. This review focuses on rigid body motion correction of head and brain MRI and its application in diagnosis and research. It explains the sources and types of motion and related artefacts, classifies and describes existing techniques for motion detection, compensation and correction and lists established and experimental approaches. Retrospective motion correction modifies the MR image data during the reconstruction, while prospective motion correction performs an adaptive update of the data acquisition. Differences, benefits and drawbacks of different motion correction methods are discussed.