Information generated by abstract interpreters has long been used to perform program specialization. Additionally, if the abstract interpreter generates a multivariant analysis, it is also possible to perform multiple specialization. Information about values of variables is propagated by simulating program execution and performing fixpoint computations for recursive calls. In contrast, traditional partial evaluators (mainly) use unfolding for both propagating values of variables and transforming the program. It is known that abstract interpretation is a better technique for propagating success values than unfolding. However, the program transformations induced by unfolding may lead to important optimizations which are not directly achievable in the existing frameworks for multiple specialization based on abstract interpretation. The aim of this work is to devise a specialization framework which integrates the better information propagation of abstract interpretation with the powerful program transformations performed by partial evaluation, and which can be implemented via small modifications to existing generic abstract interpreters. With this aim, we will relate top-down abstract interpretation with traditional concepts in partial evaluation and sketch how the sophisticated techniques developed for controlling partial evaluation can be adapted to the proposed specialization framework. We conclude that there can be both practical and conceptual advantages in the proposed integration of partial evaluation and abstract interpretation.