The advances in deep learning have revolutionized the field of artificial intelligence, demonstrating the ability to create autonomous systems with a high level of understanding of the environments where they operate. These advances, as well as new tasks and requirements in space exploration, have led to an increased interest in these deep learning methods among space scientists and practitioners. The problems of controlling the attitude and relative motion of spacecraft are considered for both traditional and new missions, such as contactless space debris removal. Both supervised and reinforcement learning is used to solve such problems based on various architectures of artificial neural networks, including convolutional ones. The possibility of using deep learning together with methods of control theory is analyzed to solve the considered problems more efficiently. The difficulties that limit the application of these methods for space applications are highlighted. The necessary research directions for solving these problems are indicated.