Due to the reduction of technological costs and the increase of satellite launches, satellite images are becoming more popular and easier to obtain. Besides serving benevolent purposes, satellite data can also be used for malicious reasons such as misinformation. As a matter of fact, satellite images can be easily manipulated relying on general image editing tools. Moreover, with the surge of Deep Neural Networks (DNNs) that can generate realistic synthetic imagery belonging to various domains, additional threats related to the diffusion of synthetically generated satellite images are emerging. In this paper, we review the State of the Art (SOTA) on the generation and manipulation of satellite images. In particular, we focus on both the generation of synthetic satellite imagery from scratch, and the semantic manipulation of satellite images by means of image-transfer technologies, including the transformation of images obtained from one type of sensor to another one. We also describe forensic detection techniques that have been researched so far to classify and detect synthetic image forgeries. While we focus mostly on forensic techniques explicitly tailored to the detection of AI-generated synthetic contents, we also review some methods designed for general splicing detection, which can in principle also be used to spot AI manipulate images.