We introduce a radical new approach to endowing neural networks with access to long-term and large-scale memory. Architecting networks with internal multigrid structure and connectivity, while distributing memory cells alongside computation throughout this topology, we observe that coherent memory subsystems emerge as a result of training. Our design both drastically differs from and is far simpler than prior efforts, such as the recently proposed Differentiable Neural Computer (DNC), which uses intricately crafted controllers to connect neural networks to external memory banks. Our hierarchical spatial organization, parameterized convolutionally, permits efficient instantiation of large-capacity memories. Our multigrid topology provides short internal routing pathways, allowing convolutional networks to efficiently approximate the behavior of fully connected networks. Such networks have an implicit capacity for internal attention; augmented with memory, they learn to read and write specific memory locations in a dynamic data-dependent manner. We demonstrate these capabilities on synthetic exploration and mapping tasks, where our network is able to self-organize and retain long-term memory for trajectories of thousands of time steps, outperforming the DNC. On tasks without any notion of spatial geometry: sorting, associative recall, question answering, our design functions as a truly generic memory and yields excellent results.