Circadian rhythms are periodic physiological events that recur about every 24 hours. The word circadian derives from the Latin words circa "about" and dies "day". The importance of circadian rhythms is well recognized in many different organisms' survival as well as in human physiology. It was in the 1950's that scientists demonstrated the existence of an endogenous biological clock, and that the clock is temperature compensated. However, the molecular mechanism of circadian rhythms began to come clear only after the discovery of the period (per) gene in Drosophila melanogaster in 1971, and the frequency (frq) gene in Neurospora crassa in 1973. Since the breakthrough discoveries of the per and frq genes and their mutants (short period mutants, perS or frq1, frq2; and long period mutants perL or frq3, frq7), molecular biologists have discovered other crucial components of the mechanism of circadian rhythms. Currently, there are about a dozen identified circadian genes in Drosophila melanogaster. The consensus idea of the mechanism is that it involves two-interlocked feedback loops largely based on transcription-translation controls. However, based on our mathematical models and analysis, we propose that there is also an autocatalytic effect based on proteolysis and stabilization of PER proteins. Based on the dynamics of multiple steady states and limit cycle oscillation, we propose an alternative mechanism for robust temperature compensation. We start with a simple model in order to understand the core dynamics of the clock mechanism, and move to a more comprehensive model. In both cases, we use bifurcation analysis as a tool to understand the dynamics of the system. With our model, we propose hypotheses to be tested in Neurospora crassa.