Existing approaches to semantics of algebraically specified data types such as Initial Algebra Semantics and Final Algebra Semantics do not take into account the possibility of general recursion and hence non-termination in the ambient programming language. Any technical development of this problem needs to be in the setting of domain theory. In this paper we present extensions of initial and final algebra semantics to algebras with an underlying domain structure. Four possibilities for specification methodologies arise: two each in the Initial and Final algebra paradigms. We demonstrate that the initial/final objects (as appropriate) exist in all four situations. The final part of the paper attempts to explicate the notion of abstractness of ADT's by defining a notion of operational semantics for ADT's, and then studying the relationship between the various algebraic-semantics proposed and the operational semantics.