It is natural to ask which properties of a smooth projective variety are recovered by its derived category. In this talk, I will consider the question: is the existence of a rational point preserved under derived equivalence? In recent joint work with Nicolas Addington, Benjamin Antieau, and Katrina Honigs, we show that over Q, the answer is no. We give two examples: an abelian variety and a torsor over it, and a pair of hyperkaehler fourfolds. The latter is independently interesting as a new example of a transcendental Brauer-Manin obstruction to the Hasse principle.