An investigation of combustion instabilities was conducted using an approximate analysis which allows any relevant physical processes to be included. The resulting system of coupled nonlinear oscillator equations was studied using the methods of dynamical systems theory. Previous investigations have further simplified the system using the method of time-averaging and truncation to a small number of modes. We have investigated the consequences of using these additional approximations, a case which had not been addressed completely in the literature. It was determined that application of the method of time-averaging introduces a stability boundary which limits the range in which the averaged equations are valid. Transverse oscillations in a cylindrical chamber were also treated. It was established that in addition to its role in energy transfer between modes, nonlinear gasdynamics also provides a means of shifting the frequencies of oscillations to integral multiples of the fundamental. This additional role can reduce the efficiency of energy transfer, thus increasing the acoustic amplitudes. An example of a low amplitude transverse oscillation was produced suggesting a means by which the amplitudes of transverse modes, as well as nonintegral longitudinal modes, may be reduced. The coupling between combustion processes and acoustic oscillations was studied as a possible explanation of the phenomenon known as triggering. Using several ad hoc models, the effects of nonlinear pressure coupling and velocity coupling on the behavior of the system were investigated. Substantial regions of possible triggering were produced when using a model of velocity coupling with a threshold, but only if nonlinear gas dynamics was also included. The interaction between combustion noise and acoustic instabilities has received relatively little attention. The sources of noise in a combustion chamber are associated with vorticity and entropy waves. By including these contributions in the approximate analysis, the general forms of the stochastic excitations were obtained. Subsequently, the efects of these excitations on the amplitudes of acoustic modes were studied. When only nonlinear gasdynamics was included, no cases of bimodal probability density functions, characteristic of triggering, were found. However, when the model of velocity coupling with a threshold is added, bimodal probability densities can occur.