We survey the asymptotic fluctuation processes for the one dimensional KPZ universality class. In particular, we will describe the formulas for the transition probabilities of TASEP and the KPZ fixed point — the special scaling invariant Markov process at the centre of the class. We will also try to explain where these formulas come from, as well as the connection with dispersive partial differential equations. Based on joint papers with Matetski and Remenik
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