Magnetic resonance elastography (MRE) is a phase-contrast-based MRI imaging technique that can directly visualize and quantitatively measure propagating acoustic strain waves in tissue-like materials subjected to harmonic mechanical excitation. The data acquired allows the calculation of local quantitative values of shear modulus and the generation of images that depict tissue elasticity or stiffness. This is significant because palpation, a physical examination that assesses the stiffness of tissue, can be an effective method of detecting tumors, but is restricted to parts of the body that are accessible to the physician's hand. MRE shows promise as a potential technique for 'palpation by imaging', with possible applications in tumor detection (particularly in breast, liver, kidney and prostate), characterization of disease, and assessment of rehabilitation (particularly in muscle). We describe MRE in the context of other recent techniques for imaging elasticity, discuss the processing algorithms for elasticity reconstruction and the issues and assumptions they involve, and present recent ex vivo and in vivo results.