Schur processes, introduced by Okounkov and Reshetikhin, are random sequences of integer partitions (Young diagrams) whose transitions are given by Schur functions. They appear in connection with many models of integrable probability : longest increasing subsequences of random permutations, lozenge or domino tilings, TASEP, etc. Thanks to a mapping to free fermions, their correlation functions can be computed explicitly and evaluated in several asymptotic regimes of interest. After reviewing the basic theory, I will present some of my contributions to this topic. Based on joint works with Dan Betea, Peter Nejjar, Mirjana Vuletić and Harriet Walsh.