  The aim of this paper is to introduce new forms of the weak and Omori-Yau
maximum principles for linear operators, notably for trace type operators, and
show their usefulness, for instance, in the context of PDE's and in the theory
of hypersurfaces. In the final part of the paper we consider a large class of
non-linear operators and we show that our previous results can be appropriately
generalized to this case.
