Rational points on algebraic varieties

The local-global principle, often called Hasse principle, refers to the statement: For varieties of a suitable type (e.g. quadrics) over a number field, the necessary congruence and real conditions constitute the only obstruction to existence of a rational point (i.e. a solution of the corresponding diophantine equation with coordinates in the given number field). In 1970, Yu. I. Manin noticed that many counterexamples to the Hasse principle for existence of rational points on varieties defined over number fields could be gathered under a common roof, by combining Grothendieck's Brauer group of varieties with classical results in number theory in the spirit of Gauss's reciprocity law. I shall first review this “Brauer–Manin obstruction” to the existence of rational points – examples will be given.
Research then went into several directions:
(i) Going beyond the Brauer-Manin obstruction for rational points for some classes of varieties. One has produced simpler and simpler classes of varieties for which a new obstruction is involved.
(ii) Most importantly, proving nevertheless that for some general classes of varieties, the Brauer–Manin obstruction is the ultimate one: if it is not in the way, then there exist (many) rational points. This is conjectured to hold for arbitrary rationally connected varieties, and is an open question for curves of any genus and for K3-surfaces. I shall describe partial results which have been obtained. Some of the recent theorems build upon results in additive combinatorics.
If time allows, I shall also mention two other themes of investigation:
(iii) Brauer-Manin obstruction for existence of integral points on suitable classes of varieties.
(iv) Brauer-Manin obstruction for the following question: For a smooth, projective variety over a number field, is the gcd of the finite field extensions over which it acquires a rational point equal to one?