Inspired by the human brain's structure and function, neuromorphic computing has emerged as a promising approach for developing energy-efficient and powerful computing systems. Neuromorphic computing offers significant processing speed and power consumption advantages in aerospace applications. These two factors are crucial for real-time data analysis and decision-making. However, the harsh space environment, particularly with the presence of radiation, poses significant challenges to the reliability and performance of these computing systems. This paper comprehensively surveys the integration of radiation-resistant neuromorphic computing systems in aerospace applications. We explore the challenges posed by space radiation, review existing solutions and developments, present case studies of neuromorphic computing systems used in space applications, discuss future directions, and discuss the potential benefits of this technology in future space missions.