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I will introduce the model of "spatial" random permutations. Permutations act on a countable set of points of R^d and a weight penalizes long jumps. The main question is about the occurrence of infinite cycles. As we will see, infinite cycles occur for densities larger than a "critical density", which is given by an exact expression. I will also discuss the relation between spatial permutations and the quantum Bose gas, and the Bose-Einstein condensation. The talk is mostly based on the preprint arxiv/0711.1188 , in collaboration with Volker Betz.