In this talk,at first we give a brief introduction to the theory of motives as wellas Murres conjectures about Chow groups.Then we present some remarks on Murres conjectures for certain product varieties.In particular,for k being an algebraically closedfield of characteristic 0,we prove that if X is an elliptic modular threefold and C is an irreducible projective smooth curve over k,then Murres conjecture(B) is true for X ×C.