In this paper we introduce the concept of weighted deficiency for abstract
and pro-$p$ groups and study groups of positive weighted deficiency which
generalize Golod-Shafarevich groups. In order to study weighted deficiency we
introduce weighted versions of the notions of rank for groups and index for
subgroups and establish weighted analogues of several classical results in
combinatorial group theory, including the Schreier index formula. Two main
applications of groups of positive weighted deficiency are given.