  We present various approaches to J. Herzog's theory of generalized local
cohomology and explore its main aspects, e.g., (non-)vanishing results as well
as a general local duality theorem which extends, to a much broader class of
rings, previous results by Herzog-Zamani and Suzuki. As an application, we
establish a prescribed upper bound for the projective dimension of a module
satisfying suitable cohomological conditions, and we derive some freeness
criteria and questions of Auslander-Reiten type. Along the way, we prove a new
characterization of Cohen-Macaulay modules which truly relies on generalized
local cohomology, and in addition we introduce and study a generalization of
the notion of canonical module.
