  We define and study generalized Dedekind symbols with values in
non--necessarily commutative groups, generalizing constructions of Sh. Fukuhara
in [Fu1], [Fu2]. Basic examples of such symbols are obtained by replacing
period integrals of modular forms (cf. [Ma1], [Ma2], [Kn1], [Kn2], [ChZ]) by
iterated period integrals introduced and studied in [Ma3], [Ma4].
