This paper gives a treatment of substitution for "parametric" objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of well-known consequences of initiality, and the work on corecursion is a general formulation which allows one to specify elements of final coalgebras using systems of equations. One source of our results is the theory of hypersets, and at the end of this paper we sketch a development of that theory which calls upon the general work of this paper to a very large extent and particular facts of elementary set theory to a much smaller extent. 1 Introduction This paper has two overall goals. The first is a general theory of substitution and corecursion having to do with final coalgebras. To give an example of the kind of phenomena we have in mind, consider any set S and form the functor F on sets defined by Fa = S \Theta a \Theta a. F is defined on functions in the u...