The study of brain connectivity with functional magnetic resonance imaging (fMRI) has attracted more and more attention in brain research. This thesis aims to gain a deeper understanding of functional connectivity networks and their impact on behavior and cognition. Functional connectivity networks are dynamic and unique, serving as unique brain fingerprints. Specifically, the fMRI signals are dynamic, showing time-varying connectivity patterns even within the same experimental condition. We investigate the dynamic functional connectivity changes with brain development. In addition, functional connectomes can be used as a unique fingerprint to identify an individual from a pool of participants. We develop novel machine learning approaches to enhance the uniqueness of functional connectivity and explore the relationship between cognitive processes and functional connectivity networks. The study sheds the light on brain network mechanisms underlying development. This thesis proposes several new analytic models to extract time-varying functional connectivity from resting-state fMRI data. Herein, each model targets a specific problem in biomedical applications. Dynamic sparse connectivity patterns (dSCPs) take advantage of both matrix factorization and time-varying fMRI time series to improve the estimation power of functional connectivity and enhance the interpretability of results. Time-varying graphical lasso (TVGL) addresses the limitation of the parameter selection of the window-based approach. GICA-TVGL performs the functional connectivity analysis without prior knowledge of the human brain. In addition, while considering the neural fingerprint ability of functional connectivity, three factors are involved: population-wise contribution, common neural activities, and phase-based information.