This article examines the structure and stability of detonations in mixtures of gases and solid particles via direct numerical simulation. Cases with both reactive and inert particles are considered. First, the two-phase flow model is presented and the assumptions that it is based upon are discussed. Steady-wave structures admitted by the model are subsequently analysed. Next, the algorithm employed for the numerical simulations is described. It is a recently developed high-order shock-capturing algorithm for compressible two-phase flows. The accuracy of the algorithm has been verified through a series of code validation and numerical convergence tests, some of which are included in this article. Subsequently, numerical results for both one-dimensional and two-dimensional detonations are presented and discussed. These results show that the mass, momentum and energy transfers between the two phases result in detonation structures that are substantially different from those observed in the corresponding purely gaseous flows. The effect of certain important parameters, such as particle reactivity, particle volume fraction, and particle diameter, are examined in detail. The numerical results predict that increased particle reactivity suppresses the flow instabilities and increases the detonation velocities. It is further predicted that sufficiently high particle volume fractions can cause detonation quenching regardless of particle reactivity.