Abstract::  Let G be a linear algebraic group acting on an affine variety V, all defined over a field k. Recently arithmetic invariant theory which deals with the case that k is an arbitrary field (the geometric invariant theory often assumes that k is algebraically closed) has been considered quite carefully. For examples, this theory provides one of the key ingredients in the algebraic part of some recent landmark results in the arithmetic of elliptic and hyperelliptic curves done by M. Bhargava, B. Gross, X. Wang, ...In this talk, we present some recent results and questions concerning this research direction.