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The race towards quantum computing is spurring rapid progress in our ability to control engineered quantum many-body systems, in the process opening new frontiers for fundamental explorations of non-equilibrium physics. In this Colloquium I will give an overview of some recent theoretical and experimental developments in this direction, focusing on questions and ideas motivated by the novel ability to measure individual degrees of freedom in these many-body systems. These developments include new probes of familiar quantum dynamics (randomized measurements, "deep thermalization"), as well as genuinely new non-equilibrium phenomena (measurement-induced phases of quantum information). A through line of these developments is the increasingly active role of the experimentalist, who must supplement experimental data with classical computation in order to detect these phenomena---thus blurring the line between "matter" and "information".