This work focuses upon the relationship and differences between results from stability analysis and breakup models for detonation waves described by step models. The simplest model that was investigated, namely the Rankine-Hugoniot detonation, considers a single step, that is, a reactive shock, i.e. a discontinuity within which energy is added due to the chemical reaction. For that model, the problem description features no length or time scale so that instability is inherently multidimensional, with frequencies and growth rates scaling with a transverse wavenumber. A stability analysis is performed showing that as long as the heat release is constant along the Hugoniot line, the conditions for stability are precisely the same as for inert shocks. It is shown however that for a broad range of conditions including in ideal gases but for stiff kinetics, the single step reactive wave is susceptible to breakup and extinction. Indeed the Riemann problem associated with the jump across the wave admits then a second solution, consisting of a weaker shock moving into the unburnt fluid, an expansion wave moving in the burnt fluid, with a contact surface between these two waves, separating unburnt and burnt fluid. A more realistic step model resolves the wave thickness, assuming the kinetics are stiff enough so that the wave structure can be approximated by an initiation zone where only incipient reaction occurs, with negligible effects, followed by a much thinner zone of intense reaction, which, for length scaled by the length of the initiation zone, is then approximated by a jump with zero thickness. That model is fairly realistic in that kinetics are often stiff enough; this includes single step Arrhenius rate model for high activation energy. Historically, a stability analysis was performed by Zaidel, which resulted in an infinite set of modes with growth rates increasing with frequency. That spectrum has been the object of much controversy, with some claiming such results are unphysical hence meaningless. Still at the very least, it shows instability, and also points to an actual instability mechanism occurring on a shorter length and faster time, thus pointing to scales resolving the thinner reaction zone. But first, the question of whether the thin reaction zone might be subject to breakup arises. Thus we show that at least for mixtures following the ideal gas model, that is indeed the case. Furthermore, these results are indeed consistent with a linear stability analysis resolving the thin reaction zone, which appears to invariably exhibit ii a non-oscillatory most unstable mode. To summarize, this thesis presented three new main contributions. First, it was shown using the normal mode approach that linear stability analysis for single jump reactive waves yield results identical to those for inert shocks in the literature and obtained using the Laplace transform approach, as long as the heat release is constant. Second, it was shown that breakup solutions, also for single jump, Rankine-Hugoniot detonations, exist for a broad range of fluid models, while the literature only mentions these for an ideal gas law. Finally, for detonations described by a square wave structure, it was shown that the thin reaction zone is also susceptible to breakup, which is consistent with the Zaidel paradox, and with results from linear analysis of the reaction zone itself.