Abstract: Random geometry and random permutations have been extremely active fields of research for several years. The former is characterized by the study of large planar maps and their continuum limits, i.e. the Brownian map, Liouville quantum gravity surfaces and Schramm–Loewner evolutions. The latter is characterized by the study of large uniform permutations and (more recently) of biased/pattern-avoiding permutations and their continuum limits, called permutons. These two fields have evolved completely separately until recently, when some surprising connections emerged: it is possible to reconstruct some universal permutons directly using Liouville quantum gravity surfaces and Schramm–Loewner evolutions. Our goal is to report on these new connections and explain how they are/might be helpful to prove new results in both areas of research.