Dynamics of resting-state functional magnetic resonance imaging (fMRI) provide a new window onto the organizational principles of brain function. Using state-of-the-art signal processing techniques, we extract innovation-driven co-activation patterns (iCAPs) from resting-state fMRI. The iCAPs' maps are spatially overlapping and their sustained-activity signals temporally overlapping. Decomposing resting-state fMRI using iCAPs reveals the rich spatiotemporal structure of functional components that dynamically assemble known resting-state networks. The temporal overlap between iCAPs is substantial; typically, three to four iCAPs occur simultaneously in combinations that are consistent with their behaviour profiles. In contrast to conventional connectivity analysis, which suggests a negative correlation between fluctuations in the default-mode network (DMN) and task-positive networks, we instead find evidence for two DMN-related iCAPs consisting the posterior cingulate cortex that differentially interact with the attention network. These findings demonstrate how the fMRI resting state can be functionally decomposed into spatially and temporally overlapping building blocks using iCAPs.