Biochemical oscillators generate cellular rhythms through complex interactions among genes, proteins, and metabolites, controlling essential cellular processes like signaling, motility, and development. Key design principles include delayed negative feedback, sufficient nonlinearity in reaction kinetics, and proper time-scale balancing of opposing reactions. Positive feedback can delay negative feedback signals, and oscillators are classified based on the topology of feedback loops. Examples include glycolysis, cyclic AMP production, and the horseradish peroxidase reaction, which were early recognized biochemical oscillations. Later, protein and gene regulatory networks, such as the PERIOD proteins in circadian control and the CYCLIN proteins in cell cycle control, were identified. The Repressilator, a synthetic gene circuit in bacteria, demonstrated oscillatory behavior. Oscillations require a negative feedback loop with a time delay, often created by transcription and translation delays, reaction intermediates, or dynamical hysteresis. The time delay must be sufficiently long, and reaction kinetics must be nonlinear to destabilize steady states. Proper time-scale balancing ensures oscillations. A three-component negative feedback loop, such as the PER protein feedback loop, can generate oscillations. Adding mRNA dynamics does not eliminate the need for explicit time delay. Incoherent feedback loops, where positive and negative feedback interact, can also produce oscillations. These mechanisms are found in various biological systems, including circadian rhythms, cell cycle regulation, and DNA synthesis. Oscillatory motifs include delayed negative feedback loops, amplified negative feedback loops, and incoherently amplified negative feedback loops. These are classified based on their feedback loop topologies. Oscillations can arise from bistability and nonlinearities in reaction kinetics. Chaotic behavior may also occur in complex reaction networks. Understanding these principles is crucial for studying cellular oscillations, as they underpin many physiological processes. Quantitative modeling helps elucidate the mechanisms of oscillations, and biochemical oscillators may evolve through genetic changes that destabilize steady states. Oscillations can lead to pathological conditions if homeostatic control is disrupted. This review highlights the importance of feedback loops, nonlinearity, and time-scale balancing in generating and maintaining biochemical oscillations.Biochemical oscillators generate cellular rhythms through complex interactions among genes, proteins, and metabolites, controlling essential cellular processes like signaling, motility, and development. Key design principles include delayed negative feedback, sufficient nonlinearity in reaction kinetics, and proper time-scale balancing of opposing reactions. Positive feedback can delay negative feedback signals, and oscillators are classified based on the topology of feedback loops. Examples include glycolysis, cyclic AMP production, and the horseradish peroxidase reaction, which were early recognized biochemical oscillations. Later, protein and gene regulatory networks, such as the PERIOD proteins in circadian control and the CYCLIN proteins in cell cycle control, were identified. The Repressilator, a synthetic gene circuit in bacteria, demonstrated oscillatory behavior. Oscillations require a negative feedback loop with a time delay, often created by transcription and translation delays, reaction intermediates, or dynamical hysteresis. The time delay must be sufficiently long, and reaction kinetics must be nonlinear to destabilize steady states. Proper time-scale balancing ensures oscillations. A three-component negative feedback loop, such as the PER protein feedback loop, can generate oscillations. Adding mRNA dynamics does not eliminate the need for explicit time delay. Incoherent feedback loops, where positive and negative feedback interact, can also produce oscillations. These mechanisms are found in various biological systems, including circadian rhythms, cell cycle regulation, and DNA synthesis. Oscillatory motifs include delayed negative feedback loops, amplified negative feedback loops, and incoherently amplified negative feedback loops. These are classified based on their feedback loop topologies. Oscillations can arise from bistability and nonlinearities in reaction kinetics. Chaotic behavior may also occur in complex reaction networks. Understanding these principles is crucial for studying cellular oscillations, as they underpin many physiological processes. Quantitative modeling helps elucidate the mechanisms of oscillations, and biochemical oscillators may evolve through genetic changes that destabilize steady states. Oscillations can lead to pathological conditions if homeostatic control is disrupted. This review highlights the importance of feedback loops, nonlinearity, and time-scale balancing in generating and maintaining biochemical oscillations.