After a short introduction to the swampland program and the challenges one faces in explicit tests of some conjectures, this talk will focus on the computation of periods and their application to the swampland program. A detailed understanding of periods allows to answer questions about the existence of solutions or the finiteness of the string landscape, as well as explicit constructions of models. Moreover, applications of periods outside of string compactifications are discussed, such as the computation of Feynman integrals, the tameness of QFTs and the appearance of periods in many other physical systems.