This chapter seeks to establish the existence of pluricomplex Green's functions with singularities at certain multi poles, given by arbitrary local analytic functions. The Green's function plays a central role in the study of functions of one complex variable or of two real variables. This chapter also attempts to develop a geometric/analytic approach to Monge-Ampère equations with measures on the right-hand side, where the singularities of the solution arise from blow-up constructions. Since blow-ups typically lead to degenerate Kähler forms, an essential tool in this chapter's approach is the recent existence theorems for the Dirichlet problem for complex Monge-Ampère equations with degenerate background form.