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The study of spontaneous brain activity using functional Magnetic Resonance Imaging (fMRI) is a relatively young and rapidly developing field born in the mid-nineties. So far, sufficiently solid foundations have been established, mainly in validating the neuronal origin of a significant component of observed low-frequency fluctuations in the 'resting state' fMRI signal. Nevertheless, the field is still facing several major challenges. This thesis first reviews the current state of knowledge and subsequently proceeds to present original research results that are directed towards overcoming these challenges. The first challenge stems from the indirect nature of the fMRI recordings, obscuring the interpretation in terms of the underlying neuronal activity. Two investigations related to this are presented. First, I show that increased head-movement, epiphenomenal to altered states of consciousness, can lead to spurious increases in low-frequency fluctuations in fMRI signal. This may adversely affect inferences on the underlying neurophysiological processes. Second, I demonstrate a direct electrophysiological correlate of increased synchronisation of fMRI activity in areas of the much studied default-mode network. By directly studying electrophysiological correlates of fMRI-based functional connectivity, this study took a pioneering approach to confirming the biological validity of the fMRI functional connectivity concept. Another widely debated question within the field is the optimal method for extracting relevant information from the extreme volumes of neuroimaging data. I present an investigation providing insights and practical recommendations for this question, based on assessing the interdependence information neglected by the commonly used linear correlation for fMRI functional connectivity studies. The results suggest that in typical resting state data, the nonlinear contributions to instantaneous connectivity are negligible. The third major challenge of the field is the integration of the experimental evidence into theoretical models of spontaneous brain activity. In the last part of this thesis, such models are discussed in detail, focusing on the two crucial features of observed spontaneous brain activity: functional connectivity and low-frequency fluctuations. Two specific mechanisms for emergence of the latter are proposed, depending either on the local synchronisation dynamics or the regulatory action of particular neuromodulators. The thesis concludes with discussion of the questions arising from the presented results in the context of the most recent development in the wider field.