Cyclic phenomena in biology are enormously varied. The periods of biological cycles range from years, such as those of oscillating fish populations, to less than a second, such as those of neuron firing. Despite their obvious differences, the mathematical analysis of apparently disparate biological cycles is often very similar. In the present thesis, I introduce the mathematical notions required to model this kind of problems, and apply them to case studies such as plankton dynamics, predator-prey models and sleep-wake cycles. In addition, I reflect on my experiences in multidisciplinary research, and highlight some of the challenges of communication and the role of software in science.