What is the stateoftheart of the proof/counterexamples of the Hasse principle for highdimensional hypersurfaces (say 3folds or more)?
Regarding counterexamples, there is SarnakWang, and then the results of Bjorn Poonen. Regarding proofs that the Hasse principle holds, I don't know the best result over ALL global fields. Of course global function fields are $C_2$, thus the Hasse principle trivially holds over global function fields for hypersurfaces of degree $d$ in $\mathbb{P}^n$ satisfying $d^2 \leq n$, because these hypersurfaces have rational points. Edit. I should add that some experts have asked / conjectured about whether the BrauerManin obstruction is the only obstruction to the Hasse principle for every rationally connected, smooth, projective variety over a global field. Since the BrauerManin obstruction vanishes for smooth hypersurfaces of dimension $\geq 3$, in particular these people are asking whether the Hasse principle holds for hypersurfaces of degree $d$ in $\mathbb{P}^n$ with $d\leq n$ and $n\geq 4$. To my knowledge there are no known counterexamples. 


A recent positive result on cubic hypersurfaces: 

