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The nervous systems converts the physical quantities sensed by its primary receptors into trains of events that are then processed in the brain. The unmatched efficiency in information processing has long inspired engineers to seek brain-like approaches to sensing and signal processing. The key principle pursued in neuromorphic sensing is to shed the traditional approach of periodic sampling in favor of an event-driven scheme that mimicks sampling as it occurs in the nervous system, where events are preferably emitted upon the change of the sensed stimulus. In this paper we highlight the advantages and challenges of event-based sensing and signal processing in the visual, auditory and olfactory domains. We also provide a survey of the literature covering neuromorphic sensing and signal processing in all three modalities. Our aim is to facilitate research in event-based sensing and signal processing by providing a comprehensive overview of the research performed previously as well as highlighting conceptual advantages, current progress and future challenges in the field.