In this contribution we propose a general presentation of Eulerian multi-fluid modeling and numerical methods for the simulation of polydisperse evaporating sprays. By spray, we denote a cloud of spherical liquid droplets of various sizes ranging from submicronic scales up to several hundred microns which interact with the carrier gaseous phase and among themselves. We deal with sprays for which the physics of such a two-phase flow is governed at the “kinetic” level, also called mesoscopic level, by a Williams-Boltzmann spray equation, where the elementary phenomena such as evaporation, heating, coalescence and secondary break-up can be described properly. Our ob jective is to provide a hierarchy of models of Eulerian type with two main criteria : 1- to take into account accurately the polydispersion of the spray, that is the large size spectrum, as well as size-conditioned dynamics, evaporation and heating, 2- to keep a rigourous link with the Williams-Boltzmann spray equation at the mesoscopic level of description. We start with the original multi-fluid model where the polydispersion is resolved by discretizing the size phase space into intervals, also called sections in relation to the original work of Greenberg and Tambour. We present the fundamentals of the model, the associated precise set of related closure assumptions as well as its implication on the mathematical structure of solutions. We provide robust numerical methods able to cope with the potential presence of singularities and a set of validations showing the efficiency of the model and of the related numerical methods. This approach is very robust but encounters two difficulties : 1- in order to accurately resolve evaporation and size conditioned dynamics in phase space, it requires a high number of size intervals, 2- for finite Stokes numbers and large Knudsen numbers, it fails to reproduce droplet crossing tra jectories since it relies on a hydrodynamic local velocity equilibrium. We thus present recent studies based on high order moment methods which allow to overcome these two difficulties. Then, the models are validated by comparisons with experimental measurements in the configuration of pulsated free jets with polydisperse spray injection, a dedicated well-controled environment with coupled Laser diagnostics. We conclude this lecture with a chapter on the computational multi-fluid dynamics and prove that the proposed models and related numerical methods and algorithms are well-suited for 2D and 3D simulations. They prove to be very accurate versus a Lagragian approach and eventually involve a small amount of numerical diffusion as a consequence of a precise choice of algorithms. Their ability for high performance computing on parallel architecture with dedicated algorithms is also demonstrated. In the framework of this Lecture Series, it provides a set of very interesting tools for the purpose of simulating nanoparticles in fluid flows.