Neurological disorders are one of the most important public health concerns in developed countries. Established brain imaging techniques such as magnetic resonance imaging (MRI) and x-ray computerised tomography (CT) have been essential in the identification and diagnosis of a wide range of disorders, although usually are insufficient in sensitivity for detecting subtle pathological alterations to the brain prior to the onset of clinical symptoms—at a time when prognosis for treatment is more favourable. The mechanical properties of biological tissue provide information related to the strength and integrity of the cellular microstructure. In recent years, mechanical properties of the brain have been visualised and measured non-invasively with magnetic resonance elastography (MRE), a particularly sensitive medical imaging technique that may increase the potential for early diagnosis. This review begins with an introduction to the various methods used for the acquisition and analysis of MRE data. A systematic literature search is then conducted to identify studies that have specifically utilised MRE to investigate the human brain. Through the conversion of MRE-derived measurements to shear stiffness (kPa) and, where possible, the loss tangent (rad), a summary of results for global brain tissue and grey and white matter across studies is provided for healthy participants, as potential baseline values to be used in future clinical investigations. In addition, the extent to which MRE has revealed significant alterations to the brain in patients with neurological disorders is assessed and discussed in terms of known pathophysiology. The review concludes by predicting the trends for future MRE research and applications in neuroscience.