This work is at the meeting point of two theories: the Theory of Hopf Algebras and the Theory of Polynomial Identities. Hopf algebras are among the most sophisticated objects of mathematics. Their theory has connections with Topology, Lie Theory, Quantum Physics and other areas. Polynomial identities are the most universal inherent properties of various algebraic systems. The theory of polynomial identities of algebras is a well-developed area. Among the classical results, there is the classification of groups whose group algebra satisfies a nontrivial identity and of Lie algebras whose universal or restricted envelope satisfies a nontrivial identity. The present work extends these results to cocommutative Hopf algebras. The notion of polynomial identity for coalgebras, which are objects dual to algebras, is new, and the theory of identities of coalgebras is not sufficiently developed. The present work includes general results in this area that are parallel to the theory of identities of algebras, and also points out some important distinctions between the two theories.

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