The Brauer group and the Chow group of $0$-cycles are two important invariants associated with an algebraic variety. For smooth varieties over local fields, these groups are related by a bilinear pairing known as the Brauer–Manin pairing. A natural question is whether this pairing is perfect, and if not, how we can describe its failure. We will begin the talk by defining this pairing and recalling some known results in dimension one, alongside a few established results for smooth surfaces. I will discuss my previous contributions to the case of curves, and finally, I will present some new results for surfaces obtained in an ongoing work with Amalendu Krishna.