This article reviews some of the intrinsic self-healing epoxy materials that have been investigated throughout the course of the last twenty years. Emphasis is placed on those formulations suitable for the design of high-performance composites to be employed in the aerospace field. A brief introduction is given on the advantages of intrinsic self-healing polymers over extrinsic counterparts and of epoxies over other thermosetting systems. After a general description of the testing procedures adopted for the evaluation of the healing efficiency and the required features for a smooth implementation of such materials in the industry, different self-healing mechanisms, arising from either physical or chemical interactions, are detailed. The presented formulations are critically reviewed, comparing major strengths and weaknesses of their healing mechanisms, underlining the inherent structural polymer properties that may affect the healing phenomena. As many self-healing chemistries already provide the fundamental aspects for recyclability and reprocessability of thermosets, which have been historically thought as a critical issue, perspective trends of a circular economy for self-healing polymers are discussed along with their possible advances and challenges. This may open up the opportunity for a totally reconfigured landscape in composite manufacturing, with the net benefits of overall cost reduction and less waste. Some general drawbacks are also laid out along with some potential countermeasures to overcome or limit their impact. Finally, present and future applications in the aviation and space fields are portrayed.