This paper studies fundamental questions concerning category-theoretic models of induction and recursion . We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor . For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras . We present a new more general proof of Taylor ’ s General Recursion Theorem that every well-founded coalgebra is recursive, and we study conditions which imply the converse . In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.