We are interested in the algebraic structure of infinite groups, possibly endowed with a non-discrete topology. The groups under study often act naturally actions on some geometric space. In addition to the classical framework of Lie groups, semi-simple algebraic groups and their discrete subgroups, generalizations are considered, notably to the Kac–Moody groups. These particular examples are brought in perspective in a more global analysis of topological groups, for which the central questions are rigidity, linearity, finiteness properties, and the existence and the classification of networks.