Abstract: There is a fruitful analogy between random regular graphs and random hyperbolic surfaces (of which there are several different models: random covers, random gluings of ideal triangles, and Weil-Petersson measure). The similarities become apparent when the number of vertices, respectively genus, goes to infinity. I will discuss counting closed geodesics on these objects, in particular the "birthday problem", which asks for the length scale at which closed geodesics become very likely to self-intersect. We solve this problem for the graph case, and discuss progress for hyperbolic surfaces. Based on joint work with Jenya Sapir.