In this talk, we introduce Brauer–Manin obstructions to the Hasse principle and to weak/strong approximation of adelic points on varieties over a global field. We also define variants of these obstructions well-suited for the study of rational points on homogeneous spaces of affine algebraic groups.

We then state the main questions considered in this theory and give an account of some developments, including the speaker’s recent work. Finally, we present briefly the main tools and ideas used in the theory, such as arithmetic duality theorems, highlighting the difficulties which appear in positive characteristic.