Inspired from the recent developments in theories of non-wellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and non-wellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the non-wellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.