It is an important conjecture that the Chow group of a smooth projective variety over a number field is a finitely generated Z-module, and no one knows how to solve it so far. On the other hand people considered two modified questions which are more accessible. One is whether the Chow group modulo n is finite or not, and the other is whether the torsion part is finite or not.

In this talk I survey recent progress about these two questions made by Shuji Saito, Kanetomo Sato and the speaker