I will give a brief introduction to the topic of random loop models, which includes random permutations coming from the interchange process (or random transpositions process) on various graphs. One of the motivations to study these models comes from statistical physics and the appearance of macroscopic loops corresponds to a phase transition in quantum Heisenberg ferromagnet or antiferromagnet. So far most of the known results are limited to graphs with rather simple geometry and the topic offers numerous open problems. Based on joint work with R. Adamczak, J. Björnberg, B. Lees, P. Miłoŝ.