How many simple closed curves can you draw on a genus g surface so that no two intersect or are homotopic? How about if you allow for curves to intersect once, but no two curves can intersect more than once? More than k times? How many times must a pair of simple closed curves intersect before they cut a genus g surface up into disks? How many different ways are there, as a function of genus, to cut a surface along a pair of simple closed curves to obtain a single disk? How about k disks? Some of these questions are elementary and complete solutions have been known for some time, while others are surprisingly difficult and remain open. In this talk we’ll discuss how studying counting problems on surfaces can quickly lead to deep and interesting questions relating to mapping class groups, the Teichmuller and moduli spaces of hyperbolic surfaces, and the topology of 3-manifolds. In principle this should be self-contained; I won’t assume background knowledge about any of this. Some of this is joint work with S. Huang.