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Measured Group Theory (MGT) studies groups through their measurable actions. It investigates classification, rigidity and invariants of (actions of) discrete countable or locally compact groups and the ways these are encoded in the orbit structures of the actions. The subject started with Gromov who suggested the notion of measured equivalence as a common framework to relate and better understand the family of (cocompact and non-cocompact) lattices in a fixed Lie group. In the last 20 years, MGT has been developed intensely. It has also established new and sometimes quite unexpected connections between various areas, including geometric group theory, von Neumann algebras, asymptotic group theory, descriptive set theory, percolation on graphs and graph convergence.