The human brain is a complex network; an intricately woven tapestry of neuronal and non-neuronal cells in constant communication with each other. Precisely how the anatomical wiring of the human brain gives rise to a repertoire of complex functions remains incompletely understood. To elucidate this intricate mapping, we employ tools from network neuroscience and control theory: the former allows us to quantify the structural and functional connectivity patterns across different brain regions, while the latter tells us how to drive the brain from an initial pattern of functional activation to a target pattern, using external input. First, to understand how flexible cognition emerges from the underlying anatomical connectivity of the brain, we analyze neuroimaging data acquired from humans and macaque monkeys and show that flexible cognition depends on the distribution of the underlying anatomical projections and the energetic costs required to switch between different patterns of functional activation. Secondly, to address how the relationship between structure and function varies across different brain regions and among individuals, we introduce the concept of structure-function coupling, a metric that quantifies how strongly a brain region’s functional expression patterns reflect the underlying structural connectivity. We review studies assessing the heterogeneous expression of structure-function coupling across brain regions, individuals, cognitive tasks, and over time, and its role in fostering flexible cognition. From a clinical perspective, we further collate studies showcasing how structure-function coupling becomes aberrant in the presence of neurological and psychiatric disorders. To investigate why structure-function coupling changes across the aforementioned dimensions, we draw insight from the fields of neurobiology and computational neuroscience. We then empirically demonstrate how different neurobiological properties operating at different timescales synergistically shape structure-function coupling across the cortex, in a region-dependent manner. Lastly, we investigate how the interplay between structure and function changes as humans transition between wakefulness and unconsciousness, and demonstrate that induction to and emergence from anesthetically-induced unconsciousness are characterized by widespread connectivity and hemodynamic changes. Collectively, this work deepens our understanding on how the brain’s anatomical architecture shapes its complex functional expression in health, in the presence of neuropsychiatric disorders, and across conscious-unconscious transitions.