In Number Theory, congruences modulo prime powers provide local information and, sometimes (e.g. for conics), everywhere local solvability is enough to imply global solvability. It's been known for some time that this does not hold for more general curves. Some recent conjectures say that a modified local-global principle for curves holds if we consider unramified covers. This talk will explain these conjectures, some consequences and recent progress. If time permits I'll also discuss a dynamical analog.