New tools are presented for reasoning about properties of recursively 
defined domains.  We work within a general, category-theoretic framework 
for various notions of `relation' on domains and for actions of domain 
constructors on relations.  Freyd's analysis of recursive types in terms of 
a property of mixed initiality/finality is transferred to a corresponding 
property of <i>invariant</i> relations.  The existence of invariant 
relations is proved under completeness assumptions about the notion of 
relation.  We show how this leads to simpler proofs of the computational 
adequacy of denotational semantics for functional programming languages 
with user-declared datatypes.  We show how the initiality/finality property 
of invariant relations can be specialized to yield an induction principle 
for admissible subsets of recursively defined domains, generalizing the 
principle of structural induction for inductively defined sets.  We also 
show how the initiality/finality property gives rise to the co-induction 
principle studied by the author (Theoretical Computer Science 124(1994) 
195-219), by which equalities between elements of recursively defined 
domains may be proved via an appropriate notion of `bisimulation'.