Various imaging and electrophysiological studies in a number of different species and brain regions have revealed that neuronal dynamics associated with diverse behavioral patterns and cognitive tasks take on a sequence-like structure, even when encoding stationary concepts. These neuronal sequences are characterized by robust and reproducible spatiotemporal activation patterns. This suggests that the role of neuronal sequences may be much more fundamental for brain function than is commonly believed. Furthermore, the idea that the brain is not simply a passive observer but an active predictor of its sensory input, is supported by an enormous amount of evidence in fields as diverse as human ethology and physiology, besides neuroscience. Hence, a central aspect of this review is to illustrate how neuronal sequences can be understood as critical for probabilistic predictive information processing, and what dynamical principles can be used as generators of neuronal sequences. Moreover, since different lines of evidence from neuroscience and computational modeling suggest that the brain is organized in a functional hierarchy of time scales, we will also review how models based on sequence-generating principles can be embedded in such a hierarchy, to form a generative model for recognition and prediction of sensory input. We shortly introduce the Bayesian brain hypothesis as a prominent mathematical description of how online, i.e., fast, recognition, and predictions may be computed by the brain. Finally, we briefly discuss some recent advances in machine learning, where spatiotemporally structured methods (akin to neuronal sequences) and hierarchical networks have independently been developed for a wide range of tasks. We conclude that the investigation of specific dynamical and structural principles of sequential brain activity not only helps us understand how the brain processes information and generates predictions, but also informs us about neuroscientific principles potentially useful for designing more efficient artificial neuronal networks for machine learning tasks.