Understanding how to characterize quantum chaotic dynamics is a longstanding question. The universality of chaotic many-body dynamics has long been identified by random matrix theory, which led to the well-established framework of the Eigenstate Thermalization Hypothesis. In this talk, I will discuss recent developments that identify Free Probability -- a generalization of probability theory to non-commuting objects — as a unifying mathematical framework to describe correlations of chaotic many-body systems. I will show how the full version of the Eigenstate Thermalization Hypothesis, which encompasses all the correlations, can be rationalized and simplified using the language of Free Probability. This approach uncovers unexpected connections between quantum chaos and concepts in quantum information theory, such as unitary designs. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006). Sponsored by the European Research Council by the Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”