Elastography, the imaging of soft-tissue elasticity, is a rapidly expanding field with a number of viable methods currently being developed in parallel. These methods are distinguishable based on the source of their raw data (most commonly ultrasound or MRI), on the manner in which the data is collected (either quasi-static, transient dynamic, or time-harmonic), and the manner in which the motion data is converted into an elasticity image (ranging from direct processing of the measured data to large scale optimisation based inverse problems). This talk explores the relation between some of these choices and the overall goal for the resulting image, i.e. is Elastography part of the screening process, the diagnostic process, or some mixture of both. The talk will look into how the computational problem changes as the quantity and quality of the data changes, and also some of the computational modelling techniques available for processing the available motion information. Specific topics include: digital image based elastographic methods; boundary element methods; shape function geometric discretization; MR Elastography, finite element methods; more sophisticated material models; imaging results in phantom and in vivo experiments.