In the first part of this talk, we shall quickly review Noether's problem, generic Galois extensions and the arithmetic lifting property (Beckmann-Black problem), along with some relations between them and some known positive and negative answers. In the second part, we return to Noether's problem but restrict ourselves to the case when the base field is algebraically closed. David Saltman (1984) was the first to come up with an example of a group which gave a negative answer to this version of Noether's problem (by studying the unramified Brauer group). Fedor Bogomolov (1986) then studied this unramified Brauer group extensively and was able to explicitly compute it in some cases leading to some more examples where the answer was negative. We will discuss the recent work of Emmanuel Peyre (2008) where he gives a new example of a group with a negative answer to Noether's problem by studying the unramified cohomology group, but now in degree 3.