In this talk I will discuss two specific questions in the theory of homogenization. First I will talk about recent progress on qualitative homogenization of degenerate elliptic equations and related probabilistic question of invariance principle for random walk in random environment. The second is more quantitative, and is related to the generalization of the multipole expansion to the case of a uniformly elliptic heterogeneous media. This is done by analyzing spaces of growing and decaying harmonic functions and relation between them.