In this paper, we describe approaches for analyzing functional MRI data to assess brain connectivity. Using phase-space embedding, bivariate embedding dimensions and delta-epsilon methods are introduced to characterize nonlinear connectivity in fMRI data. The nonlinear approaches were applied to resting state data and continuous task data and their results were compared with those obtained from the conventional approach of linear correlation. The nonlinear methods captured couplings not revealed by linear correlation and was found to be more selective in identifying true connectivity. In addition to the nonlinear methods, the concept of Granger causality was applied to infer directional information transfer among the connected brain regions. Finally, we demonstrate the utility of moving window connectivity analysis in understanding temporally evolving neural processes such as motor learning.