This talk includes two parts. The first one is devoted for a very basic introduction on the homogenization theory such as weak solutions of elliptic PDEs and two-scale asymptotic expansions, and we hope to prove the homogenization theorem for Dirichlet boundary value problems in the end of this part. In the second one, we plan to investigate some quantitative results such as convergence rates, uniform regularity estimates. In particularly, we hanve an interest in scaling-invariant estimates which is closely related to the decay estimates and the asymptoic expansions for fundamental solutions. Also, the recent developments in this field will be involved.