Rationality of Fano 3-folds over nonclosed fields

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.
In this talk I will discuss rationality  of forms of these  Fanos over nonclosed fields of characteristic 0.
I will concentrate on the case where the  Picard number  equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).