Fields of invariants for Abelian groups

The question of the rationality of fields of invariants for finite groups of transformations that act linearly on a finite-dimensional space $V$ has a long history, but still remains not fully solved. However, for abelian groups of transformations considerable progress has recently been made. The results are already close to definitive. In the present article a detailed survey of these results is given, and a number of facts are published for the first time. The presentation is made from a uniform point of view and uses regularly the techniques of algebraic tori. In this context one of the problems of Zariski is discussed.