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Treatment planning, which is a critical component of the radiotherapy workflow, is typically carried out by a medical physicist in a time-consuming trial-and-error manner. Previous studies have proposed knowledge-based or deep-learning-based methods for predicting dose distribution maps to assist medical physicists in improving the efficiency of treatment planning. However, these dose prediction methods usually fail to effectively utilize distance information between surrounding tissues and targets or organs-at-risk (OARs). Moreover, they are poor at maintaining the distribution characteristics of ray paths in the predicted dose distribution maps, resulting in a loss of valuable information. In this paper, we propose a distance-aware diffusion model (DoseDiff) for precise prediction of dose distribution. We define dose prediction as a sequence of denoising steps, wherein the predicted dose distribution map is generated with the conditions of the computed tomography (CT) image and signed distance maps (SDMs). The SDMs are obtained by distance transformation from the masks of targets or OARs, which provide the distance from each pixel in the image to the outline of the targets or OARs. We further propose a multi-encoder and multi-scale fusion network (MMFNet) that incorporates multi-scale and transformer-based fusion modules to enhance information fusion between the CT image and SDMs at the feature level. We evaluate our model on two in-house datasets and a public dataset, respectively. The results demonstrate that our DoseDiff method outperforms state-of-the-art dose prediction methods in terms of both quantitative performance and visual quality.