in this talk, first i will formulate the general setting of the affine sieve and then state its fundamental theorem. the outline of its proof will be explained. in particular, we will see that the main analytical part of the argument is to find the exact condition under which a finitely generated subgroup of gl(n,q) has property tau with respect to its ``square-free principal congruence subgroups"! at the end of the talk, some open problems will be presented.