Free probabiility theory was invented by Dan Voiculescu as a non-commutative analogue of classical probability theory, in order to understand the structure of specific non-commutative operator algebras from a kind of non-commutative probabilistic perspective. Later, Voiculescu discovered that free probability also describes the large N asymptotics of many random matrices. As large random matrices are becoming more and more prominent in many different subjects (like statistical physics, wireless communications, financial mathematics, quantum gravity, machine learning, quantum information) free probability has since its beginnings reached out to many different communities. Quite recently, new interesting connections with many-body quantum systems -- for example with the eigenstate thermalization hypothesis (ETH) or also the quantum symmetric simple exclusion process (QSSEP) -- have been observed. There, in particular, the combinatorial structure of free probabiliy, which is governed by non-crossing partitions and free cumulants, plays an important role.