The development of electric aircrafts is becoming an important technology for achieving the goals set by the European Commission for the reduction of gases emissions by 2050 in the aeronautical transportation system. However, there is a gap between the values of specific power in commercial electric machines and the ones required for aeronautical applications. Therefore, the search for alternative materials and non-conventional designs is mandatory. One emergent solution is using superconducting machines and systems to overcome the current limits of conventional electrical machines. This work reviews the new hybrid and all-electric aircraft tendencies, complementing it with recent research on the design and development of high specific power superconducting machines. This includes the main topologies for hybrid and all-electric aircraft, with an overview of the ongoing worldwide projects of these types of aircrafts, systematizing the main characteristics of their propulsion systems. It also includes the research on superconducting machines for the purpose of high specific power, considering the impact on the redesign of aircraft systems in the electrical, cooling, and fuel source sense.