摘要：This talk is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space R^n. One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity of a capacitor in R^n. This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasi-extremal distance constant of a domain in terms of its local boundary quasiconformal reflection constant. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length of the curve family joining two disjoint continua in a domain.