The ability to generalize learned representations across significantly different visual domains, such as between real photos, clipart, paintings, and sketches, is a fundamental capacity of the human visual system. In this paper, different from most cross-domain works that utilize some (or full) source domain supervision, we approach a relatively new and very practical Unsupervised Domain Generalization (UDG) setup of having no training supervision in neither source nor target domains. Our approach is based on self-supervised learning of a Bridge Across Domains (BrAD) - an auxiliary bridge domain accompanied by a set of semantics preserving visual (image-to-image) mappings to BrAD from each of the training domains. The BrAD and mappings to it are learned jointly (end-to-end) with a contrastive self-supervised representation model that semantically aligns each of the domains to its BrAD-projection, and hence implicitly drives all the domains (seen or unseen) to semantically align to each other. In this work, we show how using an edge-regularized BrAD our approach achieves significant gains across multiple benchmarks and a range of tasks, including UDG, Few-shot UDA, and unsupervised generalization across multi-domain datasets (including generalization to unseen domains and classes).