Let C be a hyperelliptic curve defined over the rational numbers, and consider the set S of all squarefree integers d such that the quadratic twist of C by d has a rational point. In this talk we will discuss the question of whether, given a prime number p, the set S contains representatives from all congruence classes modulo p. When C has genus 0 this question can be answered using elementary number theory, but for higher genera it seems to require the use of big conjectures in arithmetic geometry.