The important concern for modern software systems is to become more cost-effective, while being versatile, flexible, resilient, dependable, energy-efficient, customisable, configurable and self-optimising when reacting to run-time changes that may occur within the system itself, its environment or requirements. One of the most promising approaches to achieving such properties is to equip software systems with self-managing capabilities using self-adaptation mechanisms. Despite recent advances in this area, one key aspect of self-adaptive systems that remains to be tackled in depth is the provision of assurances, i.e., the collection, analysis and synthesis of evidence that the system satisfies its stated functional and non-functional requirements during its operation in the presence of self-adaptation. The provision of assurances for self-adaptive systems is challenging since run-time changes introduce a high degree of uncertainty. This paper on research challenges complements previous roadmap papers on software engineering for self-adaptive systems covering a different set of topics, which are related to assurances, namely, perpetual assurances, composition and decomposition of assurances, and assurances obtained from control theory. This research challenges paper is one of the many results of the Dagstuhl Seminar 13511 on Software Engineering for Self-Adaptive Systems: Assurances which took place in December 2013.