A (planar) bicycling curve is the pair of curves traced by the wheels of a bicycle as it rides over the plane. In this talk, we will mention a few relations with bicycling curves to some other --seemingly unrelated-- types of curves: Kirchoff rods, rolling extremals, and orbits of the vortex filament flow. Then we will state/explain some conjectures about these relations, for example: a relation between the hierarchy of integrals associated to the vortex filament flow and the bicycling monodromy.