This chapter introduces the concepts of Space Nuclear Power Systems (SNPSs), describes the history and nature of these ingenious energy-generating machines. The basic principles of the Radioisotope Thermoelectric Generator (RTG) and the recently developed Stirling Radioisotope Generator (SRG) are explored and an account of their application in several extra-terrestrial missions is presented. Nuclear fission power as a promising alternative for future outer planet and extra-solar explorations is discussed. The flight safety review and launch approval processes for U.S., as well as the failures and accidents for U.S. and U.S.S.R. (Russian) nuclear powered space missions since 1961 are presented chronologically. A comprehensive probabilistic consequence analysis of all conceivable potential hazards associated with nuclear powered space flights is set out. The chapter concludes with how SNPSs must be designed with the built-in safety features to minimize accidents and to prevent radiation exposure.