In Classical Probability, a sequence of random variables is said to be exchangeable if its joint distributions are invariant under all finite permutations. Ryll-Nardzeski’s Theorem establishes that exchangeability is the same as spreadability, the a priori weaker symmetry where all subsequences of the given sequence have the same joint distributions. In the non-commutative setting, it is known that the two symmetries no longer coincide for general quantum stochastic processes. We show that under very natural hypothesis there is an extension of the Ryll-Nardzewski Theorem in the noncommutative setting which covers a wide variety of models. Furthermore we obtain an extended De Finetti’s Theorem for various models including processes based on the CAR algebra and on the infinite noncommutative torus. This talk is based on joint work in progress with Valeriano Aiello and Stefano Rossi. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page