Title: Some recent results on Noether's problem

Abstract: After introduction on Noether's problem (NP) and generic polynomials, we talk about the following results:

1. NP for Sylow subgroups of symmetric groups, 2. a construction of a generic polynomial of A_5 (genericity of A.Brumer's A_5-polynomial), 3. a versal polynomial for C_8 and extendablity of C_4-extention to C_8, 4. explicit solutions for NP for some cyclic and meta-abelian groups.