Brauer groups and rational points

A class of geometrically integral varieties defined over a number field $k$ satisfies Hasse principle if a variety in this class has a $k$-rational point as soon as it has rational points in all the completions $k_v$ of the field $k$. For example, quadrics, Severi–Brauer varieties are known to satisfy this principle. However, counterexamples to Hasse Principle are also known even in the class of rational varieties.
In 1970, Manin showed that an obstruction based on the Brauer group of varieties, now referred to as the Brauer–Manin obstruction, can often explain failures of Hasse principle. However, is the Brauer–Manin obstruction the only one for the existence of rational points? Some Conjectures have been given on it (e.g. Colliot-Thélène's conjecture). In this talk, we will explain this obstruction and some known results. We will also discuss a joint work about Colliot-Thélène's conjecture (with Derenthal and Smeets).