The central subject of this book - the theory of shift-invariant algebras - is an outgrowth of the established theory of generalized analytic functions. This general framework carries along associated subalgebras of almost periodic functions of real variables and of bounded analytic functions on the unit disc. The text describes Bourgain algebras, orthogonal measures, and primary ideals of big disc algebras. The notion of a harmonic function is extended on compact abelian groups, and corresponding Fatou-type theorems are proven. Finally, the book introduces and studies classes of inductive limits of standard uniform algebras, including Blasche algebras. This is the first monograph devoted entirely to the topic.