  We show that the ring of symmetric functions in superspace is a cocommutative
and self-dual Hopf algebra. We provide formulas for the action of the coproduct
and the antipode on various bases of that ring. We introduce the ring sQSym of
quasisymmetric functions in superspace and show that it is a Hopf algebra. We
give explicitly the product, coproduct and antipode on the basis of monomial
quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym,
the ring sNSym of noncommutative symmetric functions in superspace, has a
multiplicative basis dual to the monomial quasisymmetric functions in
superspace.
