In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about - the existence of a solution to the first eigentriplet problem; - the geometry of the principal eigenvalue problem; - the asymptotic stability of the first eigenvector with possible constructive rate of convergence. This abstract theory is motivated and illustrated by several examples of differential, intro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for - some parabolic equations in a bounded domain and in the whole space; - some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models; - the kinetic Fokker-Planck equation in the torus and in the whole space; - some mutation-selection models.