The talk will stress the methods of positive operators by giving a brief survey of the scope of applications of the Krein-Rutman theorem with some of its recent variants. It is worth noticing that the Krein-Rutman theorem (as well as its linear or nonlinear variants) extend the well-known Perron-Frobenius theorem and is a fundamental tool in the study of positive eigenvalue problems with invaluable applications in Mathematical Analysis (e.g., Partial Differential Equations, Bifurcation Theory, Optimization, Order-preserving Dynamical Systems, Stability Theory).