A central problem in arithmetic geometry is to find a general procedure that provably computes the (finite) set of rational points on a curve of general type over a number field; this is an open problem even for genus 2 curves over the rational numbers. We will survey how a combination of descent constructions and the p-adic analytic method of Chabauty and Coleman can be used to carry out this computation in many cases, and discuss an application of this approach to a large dataset of genus 2 quintic curves. Along the way, we will see a connection to the classical geometry of Kummer surfaces.