A factorization of an Abelian group G is a decomposition of G into the sum G = A + B of two subsets A and B of G. The fundamental theorem on finite Abelian groups says that each finite Abelian group can be factored into a direct sum of cyclic subgroups. This book deals with general non-subgroup factorizations focusing mainly on cyclic groups. It starts with various constructions that produce new factorizations from old ones. Then it discusses periodic and non-periodic factorizations, quasiperiodicity and factorizations of periodic subsets. The authors deal with factorizations of infinite Abelian groups and their applications in combinatorics (Ramsey numbers, Latin squares and Hadamard matrices). They also investigate connections between factorizations and codes, including error correcting codes, variable length codes and integer codes.