Channel knowledge map (CKM) is a promising technology to enable environment-aware wireless communications and sensing with greatly enhanced performance, by offering location-specific channel prior information for future wireless networks. One fundamental problem for CKM-enabled wireless systems lies in how to construct high-quality and complete CKM for all locations of interest, based on only limited and noisy on-site channel knowledge data. This problem resembles the long-standing ill-posed inverse problem, which tries to infer from a set of limited and noisy observations the cause factors that produced them. By utilizing the recent advances of solving inverse problems with learned priors using generative artificial intelligence (AI), we propose CKMDiff, a conditional diffusion model that can be applied to perform various tasks for CKM constructions such as denoising, inpainting, and super-resolution, without having to know the physical environment maps or transceiver locations. Furthermore, we propose an environment-aware data augmentation mechanism to enhance the model's ability to learn implicit relations between electromagnetic propagation patterns and spatial-geometric features. Extensive numerical results are provided based on the CKMImageNet and RadioMapSeer datasets, which demonstrate that the proposed CKMDiff achieves state-of-the-art performance, outperforming various benchmark methods.