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Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties.

Characters of covariant tensor irreducible representations of the classical groups $GL(n)$, $O(n)$ and $Sp(n)$ are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of $n$-independent, universal characters. The corresponding rings, $CharGL$, $CharO$ and $CharSp$, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.