We leverage deep features extracted from a pre-trained Vision Transformer (ViT) as dense visual descriptors. We demonstrate that such features, when extracted from a self-supervised ViT model (DINO-ViT), exhibit several striking properties: (i) the features encode powerful high level information at high spatial resolution -- i.e., capture semantic object parts at fine spatial granularity, and (ii) the encoded semantic information is shared across related, yet different object categories (i.e. super-categories). These properties allow us to design powerful dense ViT descriptors that facilitate a variety of applications, including co-segmentation, part co-segmentation and correspondences -- all achieved by applying lightweight methodologies to deep ViT features (e.g., binning / clustering). We take these applications further to the realm of inter-class tasks -- demonstrating how objects from related categories can be commonly segmented into semantic parts, under significant pose and appearance changes. Our methods, extensively evaluated qualitatively and quantitatively, achieve state-of-the-art part co-segmentation results, and competitive results with recent supervised methods trained specifically for co-segmentation and correspondences.