Quadrature-based moment methods (QBMM) are employed to solve generalized population balance equations (GPBE) and are especially useful for poly-disperse multiphase flows. Starting from a closed GPBE, the unclosed moment equations are formulated and closed using QBMM. The accuracy of the closure is controlled by the order of the moments used in QBMM. For example, poly-disperse gas–particle flows can be described by a GPBE for the particle-phase mass-velocity number distribution function (NDF). In practice, the choice of the moments used in the closure is crucial. For particles with a continuous distribution of masses (e.g., same material with different diameters), the mean velocity and granular temperature of each size can be different. Thus, in addition to size moments, the velocity moments conditioned on size are needed to approximate the NDF. Here, the particle-phase model found from the GPBE with the Boltzmann–Enskog collision operator will be used to explain the methodology. Once the moment equations have been formulated, the numerical algorithms used to solve them must be consistent with the underlying GBPE. For example, the numerical methods employed to solve the spatial advection terms and the source terms must guarantee that the transported moments remain realizable (i.e., they must correspond to a NDF). This can be accomplished with kinetic-based, finite-volume methods. With QBMM, the NDF is represented by a finite set of weighted delta functions, corresponding to discrete velocities and sizes, that agree with the transported moments. Thus, it is often convenient to develop algorithms in terms of the quadrature variables in place of the moments. Employing applications from poly-disperse gas–particle flows, several examples of the numerical issues arising with QBMM will be discussed, along with some open issues related to the numerical algorithms.