Recent advances in artificial intelligence (AI) have significantly intensified research in the geoscience and remote sensing (RS) field. AI algorithms, especially deep learning-based ones, have been developed and applied widely to RS data analysis. The successful application of AI covers almost all aspects of Earth observation (EO) missions, from low-level vision tasks like super-resolution, denoising and inpainting, to high-level vision tasks like scene classification, object detection and semantic segmentation. While AI techniques enable researchers to observe and understand the Earth more accurately, the vulnerability and uncertainty of AI models deserve further attention, considering that many geoscience and RS tasks are highly safety-critical. This paper reviews the current development of AI security in the geoscience and RS field, covering the following five important aspects: adversarial attack, backdoor attack, federated learning, uncertainty and explainability. Moreover, the potential opportunities and trends are discussed to provide insights for future research. To the best of the authors' knowledge, this paper is the first attempt to provide a systematic review of AI security-related research in the geoscience and RS community. Available code and datasets are also listed in the paper to move this vibrant field of research forward.