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In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit of uniform meanders and some recent results on a related model called meandric systems. A meandric system is a coupled collection of meanders. Also in this case, I will present (1) a conjecture which describes the large-scale geometry of a uniform meandric system and (2) several rigorous results which are consistent with this conjecture. Based on joint works with Ewain Gwynne and Xin Sun, and Ewain Gwynne and Minjae Park.