SUMMARYPhysiologists both admire and fear complexity, but we have made relatively few attempts to understand it. Inherently complex systems are more difficult to study and less predictable. However, a deeper understanding of physiological systems can be achieved by modifying experimental design and analysis to account for complexity. We begin this essay with a tour of some mathematical views of complexity. After briefly exploring chaotic systems,information theory and emergent behavior, we reluctantly conclude that, while a mathematical view of complexity provides useful perspectives and some narrowly focused tools, there are too few generally practical take-home messages for physiologists studying complex systems. Consequently, we attempt to provide guidelines as to how complex systems might be best approached by physiologists. After describing complexity based on the sum of a physiological system's structures and processes, we highlight increasingly refined approaches based on the pattern of interactions between structures and processes. We then provide a series of examples illustrating how appreciating physiological complexity can improve physiological research, including choosing experimental models, guiding data collection, improving data interpretations and constructing more rigorous system models. Finally, we conclude with an invitation for physiologists, applied mathematicians and physicists to collaborate on describing, studying and learning from studies of physiological complexity.