Human brain systems exhibit complex organizational properties at multiple scales. Although these properties may manifest signatures of nonlinear dependence among non-invasively recorded functional neuroimaging signals, functional connectivity analyses of human functional magnetic resonance imaging (fMRI) data have typically deployed linear dependence estimators. Linear functional connectivity analyses have provided valuable insights into the organization of large-scale brain systems. However, linear estimators cannot, even in principle, detect the presence of nonlinear relationships. Therefore, prospectively fundamental aspects of nonlinear network organization at fMRI scales of observation have remained unelucidated. This dissertation aims to address that gap in knowledge by proposing a method for the data-driven extraction of brain networks from explicitly nonlinear whole-brain functional connectivity patterns. By disentangling nonlinear from linear functional connectivity information at the global level and subsequently implementing an independent component analysis blind source separation approach for learning network structure, the proposed method isolates voxel-resolution signatures of nonlinear brain ensembles. To evaluate the potential of the method to provide insights within key investigative domains, we deployed it within clinical and developmental contexts. Applied to a multi-site resting-state fMRI psychosis dataset, we found that explicitly nonlinear networks exhibit unique spatial distributions across the brain relative to their linear counterparts, leading to a significantly enhanced ability to distinguish individuals with schizophrenia from controls. Moreover, we found that the spatiotemporally dynamic properties of many explicitly nonlinear networks are significantly associated with performance in multiple cognitive domains and with schizophrenia diagnosis, symptomology, and polygenic risk. When applied to a cohort of typically developing human infants, our approach revealed that explicitly nonlinear fMRI ensembles are present as early as birth and reveal flexible trajectories of postnatal brain maturation which would be missed by conventional linear approaches. Our findings therefore reframe early functional brain reorganization in terms of dynamic, nonlinear, spatially varying coordination across distributed systems. Overall, the work of this dissertation positions nonlinear intrinsic connectivity networks as a critical source of information for characterizing and refining models of large-scale functional brain architecture.