The modeling and simulation of multiphase reacting flows covers a large spectrum of applications ranging from combustion in automobile and aeronautical engines to atmospheric pollution as well as biomedical engineering or pyroclastic flows. In the framework of this seminar, we will mainly focus on a disperse liquid phase carried by a gaseous flow field which can be either laminar or turbulent; however, this spray can be polydisperse, that is constituted of droplets with a large size spectrum; we have to deal with coalescence, break-up, droplet trajectory crossing and reduced order model for turbulent flows. Thus, such flows involve a large range of temporal and spatial scales, which have to be resolved in order to capture the dynamics of the phenomena and provide reliable and eventually predictive simulation tools. Even if the power of the computer resources regularly increases, such very stiff problems can lead to serious numerical difficulties and prevent efficient multi-dimensional simulations. The purpose of the presentation is to introduce to the Eulerian modeling of polydisperse evaporating spray for various applications, that is the disperse liquid phase carried by a gaseous flow field is modeled by "fluid" conservation equations. Such an approach is very competitive for real applications since it has strong ability for optimization on parallel architectures and leads to an easy coupling with the gaseous flow field resolution. We will show that all the necessary steps in order to develop a new generation of computational code have to be designed at the same time with a high level of coherence: mathematical modeling through Eulerian moment methods, development of new dedicated stable and accurate numerical methods, implementation of optimized algorithms as well as verification and validations of both model and methods using other codes and experimental measurements.