In this talk, we consider non-homogeneous, second order, uniformly elliptic systems of partial differential equations. We show that, within a suitable framework, we can define the fundamental solution and the Green functions on arbitrary open subsets. Moreover, we can prove uniqueness and global estimates that are on par with those of the underlying homogeneous elliptic operator. Our results, in particular, establish the Green functions for Schrodinger, magnetic Schrodinger, and generalized Schrodinger operators with real or complex coefficients on arbitrary domains.