Detonation waves are supersonic combustion waves that have a complex three-dimensional cellular structure. There is growing experimental evidence that the cellular structure of detonations promotes their propagation in the presence of losses. In spite of that, the conventional model for the detonation structure, known as the Zeldovich - Von Neumann - Doring (ZND) model, neglects the existence of cellular structure for detonations and assumes the wave to consist of a strong leading planar shock coupled with trailing chemical reactions. Therefore, the influence of cellular structure on the dynamics and extinction limits of detonation waves has been of particular interest. Previous studies have investigated the influence of cellular structure on the dynamics of detonations with mass divergence in the framework of narrow tubes, porous-walled tubes and weak confinement. However, precise quantification of the loss mechanism in these frameworks has been associated with some difficulties. Complex flow in the boundary layers, inherent in thin tubes, or attenuation of the transverse waves in the porous-walled tubes has made the evaluation of the loss mechanism more difficult in such geometries. In this thesis, a novel well-posed problem is formulated for detonations with mass divergence. It is shown that detonations propagating in a channel with a cross-section area increasing exponentially have a constant mass divergence. The detonations were found to propagate at a quasi-steady speed below the ideal Chapman-Jouguet velocity. This permitted to make meaningful comparison with the theoretical models and simulations. The experiments were performed in two mixtures, one displaying characteristic weakly unstable detonations (2C2H2 + 5O2 + 21Ar), and the other displaying highly unstable detonations (C3H8 + 5O2). The dependence of the velocity deficits and limits on the amount of mass divergence for the two mixtures were compared with the predictions of the quasi-one-dimensional ZND model with lateral mass divergence. Since the ZND model neglects the cellular structure of the detonations, such comparison permitted to asses the influence of cellular structure on the dynamics of detonations with mass divergence. Comparisons were also made with the results of simulations of inviscid cellular detonations. These comparisons showed that the velocity deficits and critical rate of mass divergence in the weakly unstable mixture were reasonably well predicted by the quasi-one-dimensional model. For smaller values of mass divergence rate, a good agreement between the experiments and the predictions of the two-dimensional cellular simulations was observed for the weakly unstable mixture. For the highly unstable detonations, the quasi-one-dimensional model significantly over-predicted the effect of mass divergence.Detonations were observed for rates of mass divergence 93% higher than the critical predicted value, displaying more substantial velocity deficits than predicted. Such observations show conclusively that the ZND model cannot capture the dynamics of highly unstable detonations on large scales.