Elastography is a useful and interesting technique that can be used to infer stiffness information from ultrasound medical images. In one decade of activity, the scientific community has developed this technique to a more and more mature stage, such that it has evolved into a fruitful application for clinical practice. During this decade, the evolution has proceeded from qualitative stiffness information to numerical quantification using different algorithms and approaches, based on both iterative and direct approaches. Moreover, different post- and pre-processing techniques as well as evaluation methods have been implemented. This work presents a survey on the methods developed by giving a short overview of algorithms and mathematical solutions and by analysing results and comparing methodologies.