From the initial discovery of free-running endogenous circadian oscillations of Crassulacean acid metabolism (CAM) under constant conditions in the light and in air, it has been disputed whether the underlying oscillator is enzymic or biophysical. The hypothesis of a biophysical hysteresis switch or beat oscillator started from osmotic considerations of malate accumulation and remobilisation, indicating a tonoplast tension/relaxation mechanism. It then advanced to application of non-linear dynamics theory for the analysis of rhythmic and arrhythmic time series of CO2 exchange under the regime of external control parameters, mainly temperature, and the implementation of models for computer simulations of CAM rhythms. This provided strong evidence for the tonoplast functioning as a master switch for circadian regulation of CAM. Conversely, the hypothesis of an enzymic beat oscillator strongly developed on the experimental basis of phosphorylation/dephosphorylation of phosphoenolpyruvate carboxylase (PEPC) regulating the enzyme activity, and hence CO2 fixation and malate synthesis via this enzyme. It was much supported by the discovery that PEPC-kinase gene-transcription was under circadian control. However, biochemical and molecular analysis, as well as model simulation, strongly suggests that this is a secondary and not the primary oscillator. The synchronisation/desynchronisation of leaf patches has revealed spatiotemporal characteristics of circadian rhythmicity that may open new ways for understanding biological clocks.