Book excerpt: This book addresses the molecular bases of some of the most important biochemical rhythms known at the cellular level. Clarifying the mechanism of these oscillatory phenomena is of key importance for understanding the origin as well as the physiological function of these rhythms, and the conditions in which simple periodic behaviour transforms into complex oscillations, including bursting and chaos. The approach rests on the analysis of theoretical models closely related to experimental observations. Among the main rhythms considered are glycolytic oscillations observed in yeast and muscle, oscillations of cyclic AMP in Dictyostelium amoebae, intracellular calcium oscillations observed in a variety of cell types, the mitotic oscillator that drives the cell division cycle in eukaryotes, and circadian oscillations of the period protein (PER) in Drosophila. For each of these phenomena, experimental facts are reviewed and mathematical models presented.