The Borromean rings are three interlinked circles such that no two circles are linked: if we cut or remove one of the circles, the other two fall apart. Massey products are an algebraic manifestation of this phenomenon. Born as part of Algebraic Topology, they have now made a surprising appearance in Number Theory and Galois Cohomology. The Massey Vanishing Conjecture of Minac and Tan predicts that all Masseyproducts in the Galois cohomology of a field vanish as soon as they are defined. In this talk, I will give an informal introduction to Massey products in Topology and Galois Theory, and then describe recent progress on the Massey Vanishing Conjecture, joint with Alexander Merkurjev