One of the most difficult challenges facing the development of modern gas turbines-for power generation, and propulsion-is the mitigation of dynamic instabilities in the presence of efficiency and emissions constraints. Dynamic instabilities-self-excited, self-sustaining oscillations which link the combustor acoustics to the combustion process-can result in significant levels of thermal and mechanical stress on combustion systems, leading to reduced operational lifetime, potentially dangerous failure modes, and significant deviations from the desired operating conditions. Due to the complexity of the problem, with the relevant time and length scales of the system--from the chemistry to the acoustics-spanning several orders of magnitude, even sophisticated numerical techniques have been severely limited in their ability to make reliable predictions, leaving the task of finding and eliminating modes of instability to a lengthy and expensive trial-and-error process. Lean-premixed combustion, one of the leading technologies for low emission combustors, is particularly susceptible to these types of instabilities. The sealed systems that are necessary to maintain a reaction in a lean mixture do not attenuate acoustics well, which often results in high-amplitude pressure fluctuations. In this thesis, we focus on developing a better predictive framework for the onset of combustion instabilities in a swirl-stabilized, lean-premixed combustor. We correlate the self-excited acoustic behavior with quantifiable system properties that can be generalized across different fuel blends. This work is predicated on the idea that self-excited combustion instability arises from the selective amplification of the noise inherent in a turbulent combustion system, and that the frequency-based response of the flame is a function of the flame geometry. In the first part of the thesis, we focus on the flame geometry, identifying several discrete transitions that take place in the swirl-stabilized flame as we adjust the equivalence ratio. By comparing the transitions across several CH4/H2 fuel blends, and using statistical techniques to interrogate the global effect of the small-scale flow-flame interactions, we find that the extinction strain rate-the flow-driven rate of change in flame surface area at which the chemistry is no longer -sufficiently fast to maintain the reaction-is directly linked to the flame transitions. The swirl-stabilized flow features several critical regions with large and unsteady velocity derivatives, particularly, a pair of shear layers that divide the incoming flow of reactants from an inner and an outer recirculation zone. As the extinction strain rate increases with increasing equivalence ratio, the flame transitions through these critical regions, manifesting as discrete changes in the flame geometry. In the second part, we address the correlation between self-excited instability and the forced acoustic response. By modifying the pressure boundary conditions, we decouple the flame from the acoustics over a domain of interest (defined by a range of equivalence ratios that correspond to the onset of dynamic instability in the coupled system). We then apply external acoustic forcing at a single frequency to ascertain the response of the flame to each particular forcing frequency by means of a flame transfer function. This enables us to consider the frequency-by-frequency response of the flame to its own internally generated noise. We show that the onset of instability is well-predicted by the overlap of the natural acoustic frequencies of the combustor (predicted using a non-linear flame response model) with those frequencies for which the phase of the flame transfer function satisfies the well-known Rayleigh criterion, which is a necessary condition for the presence of self-excited combustion instability. By examining both the forced response and the self-excited instability across several different fuel blends, we go on to show that both behaviors correlate well with the flame geometry, which we have already shown to be dictated by the extinction strain rate of the particular fuel blend. We go on to collapse both sets of data on the strained flame consumption speed taken at the limit of the extinction strain rate, and in doing so, present a framework for predicting the operating conditions under which the combustor in the coupled configuration will go unstable based on measurements and correlations from the uncoupled configuration. Furthermore by taking the consumption speed at the extinction limit, we are correlating the geometry and dynamics with a parameter that is solely a function of mixture properties. This provides the basis for a framework for predicting instability from properties that are more readily measured or simulated, and provides and explicit means of converting these results to different fuel mixtures.