In this talk, I will discuss several notions that are extensions of the principal eigenvalue of a linear elliptic operator to the framework of unbounded domains along with some of their properties. I will describe applications to semi-linear elliptic equations and to propagation in reaction-diffusion equations of the KPP type. These equations are considered in unbounded domains and for general heterogeneous settings. I report here on joint works with Luca Rossi, with François Hamel and with Grégoire Nadin.