A fundamental goal of systems neuroscience is to understand how the collective dynamics of neurons encode sensory information and guide behavior. To answer this important question one needs to uncover the network of underlying neuronal interactions. Whereas advances in technology during the last several decades made it possible to record neural activity simultaneously from a large number of network elements, these techniques do not provide information about the physical connectivity between the elements being recorded. Thus, neuroscientists are challenged to solve the inverse problem: inferring interactions between network elements from the recorded signals that arise from the network connectivity structure. Here, we review studies that address the problem of reconstructing network interactions from high-dimensional datasets generated by modern techniques, and focus on the emerging theoretical models capable of capturing the dominant network interactions of any order. These models are beginning to shed light on the structure and complexity of neuronal interactions.