Estimating correspondences between pairs of deformable shapes remains challenging. Despite substantial progress, existing methods lack broad generalization capabilities and require domain-specific training data. To address these limitations, we propose a fundamentally new approach to shape correspondence based on denoising diffusion models. In our method, a diffusion model learns to directly predict the functional map, i.e. a low-dimensional representation for a point-wise map between shapes. We use a large dataset of synthetic human meshes for training and apply two steps to reduce the number of functional maps that need to be learned. First, maps refer to a template rather than to shape pairs. Second, a functional map is defined in the basis of eigenvectors of the Laplacian, which is not unique due to sign ambiguity. We, hence, introduce an unsupervised approach to select a specific basis by correcting the signs of eigenvectors based on surface features. Our approach achieves superior performance on standard human datasets, meshes with anisotropic connectivity, and non-isometric humanoid shapes compared to existing descriptor-based and large-scale shape deformation methods. We will release the source code and the datasets for reproducibility and research purposes.