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In this work, we address the challenge of multi-domain translation, where the objective is to learn mappings between arbitrary configurations of domains within a defined set (such as $(D_1, D_2)\rightarrow{}D_3$, $D_2\rightarrow{}(D_1, D_3)$, $D_3\rightarrow{}D_1$, etc. for three domains) without the need for separate models for each specific translation configuration, enabling more efficient and flexible domain translation. We introduce Multi-Domain Diffusion (MDD), a method with dual purposes: i) reconstructing any missing views for new data objects, and ii) enabling learning in semi-supervised contexts with arbitrary supervision configurations. MDD achieves these objectives by exploiting the noise formulation of diffusion models, specifically modeling one noise level per domain. Similar to existing domain translation approaches, MDD learns the translation between any combination of domains. However, unlike prior work, our formulation inherently handles semi-supervised learning without modification by representing missing views as noise in the diffusion process. We evaluate our approach through domain translation experiments on BL3NDT, a multi-domain synthetic dataset designed for challenging semantic domain inversion, the BraTS2020 dataset, and the CelebAMask-HQ dataset.