The growing interest in Artificial Intelligence is pervading several domains of technology and robotics research. Only recently has the space community started to investigate deep learning methods and artificial neural networks for space systems. This paper aims at introducing the most relevant characteristics of these topics for spacecraft dynamics control, guidance and navigation. The most common artificial neural network architectures and the associated training methods are examined, trying to highlight the advantages and disadvantages of their employment for specific problems. In particular, the applications of artificial neural networks to system identification, control synthesis and optical navigation are reviewed and compared using quantitative and qualitative metrics. This overview presents the end-to-end deep learning frameworks for spacecraft guidance, navigation and control together with the hybrid methods in which the neural techniques are coupled with traditional algorithms to enhance their performance levels.