Artificial intelligence is applied to many fields and contributes to many important applications and research areas, such as intelligent data processing, natural language processing, autonomous vehicles, and robots. The adoption of artificial intelligence in several fields has been the subject of many research papers. Still, recently, the space sector is a field where artificial intelligence is receiving significant attention. This paper aims to survey the most relevant problems in the field of space applications solved by artificial intelligence techniques. We focus on applications related to mission design, space exploration, and Earth observation, and we provide a taxonomy of the current challenges. Moreover, we present and discuss current solutions proposed for each challenge to allow researchers to identify and compare the state of the art in this context.