Distributional symmetries and invariance principles of sequences of random variables lead to deep structural results in probability theory. For example, exchangeability means that the joint distributions of a sequence is invariant under finite permutations of the random variables. Now the classical de Finetti's theorem characterizes exchangeable infinite sequences to be conditionally i.i.d. A natural question is to ask for noncommutative versions of this fundamental result. I will report on recent progress in this matter.

A very recent result is the free version of de Finetti's theorem, obtained in joint work with Roland Speicher. Replacing the role of permutations by quantum permutations we obtain a new characterization of Voiculescu's freeness with amalgamation.

Also, I will present a more general noncommutative version of de Finetti's theorem where conditional independence emerges in terms of commuting squares of von Neumann algebras. Finally, if time permits, I will address results on braidability, a new symmetry recently introduced by Rolf Gohm and the speaker.