Understanding what deep network models capture in their learned representations is a fundamental challenge in computer vision. We present a new methodology to understanding such vision models, the Visual Concept Connectome (VCC), which discovers human interpretable concepts and their interlayer connections in a fully unsupervised manner. Our approach simultaneously reveals fine-grained concepts at a layer, connection weightings across all layers and is amendable to global analysis of network structure (e.g., branching pattern of hierarchical concept assemblies). Previous work yielded ways to extract interpretable concepts from single layers and examine their impact on classification, but did not afford multilayer concept analysis across an entire network architecture. Quantitative and qualitative empirical results show the effectiveness of VCCs in the domain of image classification.