Brauer group and rational points on K3 surfaces

I'll discuss the results on the Brauer group of K3 surfaces obtained jointly with Zarhin and Ieronymou, in particular a uniform upper bound on the cardinality of Br(X)/Br(Q) for diagonal quartic surfaces X over Q, and examples when this group is trivial. I will mention a very recent result of Swinnerton-Dyer in the direction to weak approximation for a specific diagonal quartic surface over Q.