2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106295The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of their consequences.37 pages, revised version, some recent results were addedAnalysis of PDEsSpectral Theory35J15; 35B05; 35C15; 35K10text