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Thermoacoustic instabilities are a mayor problem in industrial combustors, where they can lead to catastrophic hardware damage. An industrial gas turbine combustion chamber is a very complex and expensive system. Thus, a laboratory burner has been built for research purposes, where a large number of parameters can be varied. This study is part of the Marie Curie research network LIMOUSINE, which was set up to model thermoacoustic instabilities in the combustor chamber of gas turbines. The objective of the present thesis is to theoretically model and analyze thermoacoustic instabilities in the LIMOUSINE laboratory burner. A mathematical model of the laboratory burner has been developed. A more general form of the wave equation has been derived in the time-domain, in which the mean temperature gradient was taken into account. The governing differential equation has been solved by applying the Green’s function approach, which allows separating the effects of the unexcited burner and the fluctuating heat-release. Using perturbation techniques general solutions are given for the cases when the temperature increase is either small or large. Conclusions have been drawn about the necessary complexity of thermoacoustic models by comparing increasingly complex configurations. The forcing term of the wave equation is studied by investigating the kinematics of ducted premixed flames theoretically, and a new heat-release law is derived. Instability criterion has been derived by applying the non-linear source term. The stability parameter map of the burner has been also investigated. Expressions for the limit-cycle amplitudes and frequencies were derived using weakly non-linear theory. The predictions of the mathematical model have been compared to measurements.