This monograph provides new functional analytic developments on spectral problems in various applied fields such as jump equations, Kolmogorov differential equations, weighted graphs, neutron transport theory, population dynamics, linearized non-local Allen?Cahn equations and perturbed convolution semigroups.  With an emphasis on spectral problems (in Lebesgue spaces) connected to positivity, it uses powerful tools developed over the last two decades to unify and extend established results in various directions, offering new perspectives in a diverse range of subjects.  Many open questions are scattered throughout, in the hope of promoting further research.  Covering topics lying at the intersection between functional analysis and partial differential equations, the book is primarily targeted toward applied mathematicians working in kinetic theory, probability theory, mathematical biology and, more generally, partial differential equations, who are interested in peripheral spectral theory. It may also draw pure mathematicians working in semigroup theory towards very rich applied areas. Although mainly aimed at professional mathematicians, it will also be useful to PhD students and post-doctoral researchers.  The long chapter devoted to transport theory will be of interest to physicists or engineers involved in neutron transport.