I will consider a family of diagrammatic objects (well-known to mathematical physicists and to combinatorialists) which go under the name of “meandric systems”. I will explain how meandric systems arise in calculations done in a noncommutative probability framework, and I will show how tools from free probability can be used to study the asymptotic behaviour of a random meandric system of large order. The talk is based on a joint work with Ian Goulden and Doron Puder, and on a joint work with Ping Zhong.