Safety-critical power conversion systems play a major role in the paradigm shift towards more electric aircraft (MEA) architectures. This paper reviews the electrical machines and their power electronic systems that are currently competing in the application of integrated starter-generators (S/Gs) in MEA power systems. Motivated by the strict requirements of sufficient electrical starting capability, super-high power density and ultra-high reliability, additional considerations on the overall system design are necessary, including the power electronic converters (PECs) and integrated thermal designs. These aspects are discussed not only in the light of their many benefits but also of the challenges introduced by the continuous advancements and emerging innovations in the power conversion technology. In achieving the MEA goals and capitalize on all potential benefits, optimization-based design approaches will be necessary, where the aggregation of electric machines, PECs and the aircraft grid is considered as an integrated system to be optimized. This review highlights the importance of these aspects and offers a view on future perspectives and open issues.