Abstract: We consider the abstract Zeldovich - von Neumann - Doring (ZND) equations for combustion, which is a multidimensional balance law with abstract structural assumptions satisfied by the physical equations. The investigation of multidimensional stability of planar ZND detonation waves leads to a stability function which is a hybrid Evans function-Lopatinski determinant. Stability in the high frequency regime leads to a Lopatinski type stability condition, which is shown to be satisfied for the physical equations. This in turn allows us to construct curved multidimensional ZND detonation waves by getting an L² estimate to a suitable linearization of the resulting nonlinear initial boundary value problem, gotten with the use of the paradifferential calculus, which is good enough to solve the full nonlinear problem by Picard iteration.