I'll start with some motivating remarks (edit: as pointed out in the comments, these motivational remarks do not hold for surfaces: there is an example of a conic bundle surface over a real quadratic field with no rational points but non-empty etale-Brauer set). When it comes to obstructions to the Hasse principle or WA, it seems to me that we need to come up with a new obstruction every time we go up one dimension: it is likely that for curves (dimension 1) the Brauer-Manin obstruction (BMO) to HP or WA is, at worst, the only one; similarly, for surfaces (dimension 2), the BMO is not enough, but the descent obstruction (DO) to HP or WA might be the only one (edit: No, the DO is also not enough, as pointed out by Felipe Voloch and Martin Bright in the comments); we know that the descent obstruction to the HP for 3-folds is not enough (there is a counter-example due to Poonen), but there is probably some finer (yet) unknown obstruction which explains violations for 3-folds; and so on.
My question is on the state-of-the-art for the dimension 2 case:
Up to date, for which classes of nice surfaces do we know that the "Descent principle" (i.e. DO is, at worst, the only obstruction for HP and WA) holds?
Of course, the more general the class (e.g. rational, K3, Enriques, ...) and the number field, the better! I am already aware of some classes of surfaces for which the Descent principle holds (e.g. Chatelet surfaces over $\mathbb{Q}$, del Pezzo surfaces of degree $\geq 5$, ...), but the literature is too huge to keep track of everything. It goes without saying that references are greatly welcomed!
