This monograph is devoted to the theory of hypercomplex systems with locally compact basis. Such systems were introduced by Yu. Berezansky and S. Krein in the 1950s and are a generalization of the notion of a hypergroup (a family of generalized shift operators) which was introduced in the 1970s. The book gives an account of hypercomplex systems theory. After the introductory chapter, it treats the Lie theory of hypercomplex systems and examples. Topics covered include Fourier transforms, the Plancherel theorem, the Peter-Weyl theorem, representation theory, duality, Gelfand pairs, Sturm-Liouville operators, and Lie theory. New proofs of results concerning Tannaka-Krein duality and Gelfand pairs are given. On the basis of this theory, new approaches to the construction of harmonic analysis on well-known objects become possible.