A local system on the Riemann sphere minus finitely many points is defined to be ``rigid'' if it is determined by the conjugacy classes of its monodromy operators along the missing points. Katz proves a convenient cohomological criterion characterizing irreducible rigid local systems, which is based on an analysis of the moduli of local systems on the punctured Riemann sphere. In this talk, we will discuss this story, and then proceed towards an analogous story in the arithmetic setting, where, in place of local systems on the punctured Riemann sphere, we consider overconvergent isocrystals on the punctured projective line over a field of positive characteristic.