In a theory of processes the names are atomic data which can be exchanged and tested for identity, but which admit no other functions or predicates. A well-known example of a calculus for name-passing is the p-calculus, where names additionally are used as communication ports. We provide complete axiomatisations of late and early bisimulation equivalences in such calculi. Since neither of the equivalences is a congruence we also axiomatise the corresponing largest congruences. We consider a few variations of the signature of the language; among these, a calulus of deterministic processes which is reminiscent of sequential functional programs with a conditional construct. Most of our axioms are shown to be independent. The structure of the systems reveals the symmetries of the calculi and equivalences since they differ only by a few simple axioms.