Abstract: The aim of this talk is to give a survey of recent developments in the rationality problem for fields of invariants, e.g. Noether's problem, rationality problem for algebraic tori, rationality problem for quasi-monomial actions. I will give some explicit examples of (non-)stably (resp. retract) rational fields which are weaker concepts than the rationality. Negative results may be obtained by using (stable) birational invariants, e.g. flabby class, unramified Brauer (cohomology) group which is an avatar of the Artin-Mumford invariant H^3(X,Z)_{tors}. By Colliot-Thelene and Voisin's result (2012, Duke Math. J.), an explicit example of non-vanishing of the unramified cohomology group of degree three provides an explicit counter-example of the integral Hodge conjecture.