With current methods, the study of transcendence properties of special values of automorphic functions is really the study of periods of elliptic and abelian functions and the relations of linear dependence over the algebraic numbers that can exist between them. This is in turn an offshoot of Hilbert's seventh problem. We discuss some recent progress in this area, giving some idea of proofs, and relating it to such questions as arithmeticity of monodromy groups of hypergeomertic functions, and distribution questions for CM points on subvarities of Shimura varieties.