The aim of this talk is to present some new form of the weak (and Omori-Yau) maximum principle for general linear operators on a Riemannian manifold, notably trace operators, that make particularly clear the function theoretical aspects of this type of results and free them from the request (at least for the weak case) of completeness fo the underlying metric. We also discuss applications to gradient Ricci solitons and to the geometry of immersions to show the versatility and powerfulness of the tools when applied to geometric problems.