This chapter presents the theory of algebraic groups over locally compact fields. The first section discusses the key topological and analytic properties of the sets of rational points of algebraic varieties over local fields, and then applies these to the analysis of the groups of rational points of linear algebraic groups over such fields. The second section deals with the classical case where the field of definition is either the field of real numbers or the field of complex numbers. The third and fourth sections are devoted to algebraic groups over non-archimedean fields, and rely on techniques from the theory of profinite groups, the reduction of algebraic varieties, and Bruhat–Tits theory. The concluding section summarizes some results from measure theory that are needed for subsequent chapters.