There are basically two steps in dispersed two-phase flow modeling. The first one deals with the selection of the variables entering the particle state vector and with the construction of the stochastic processes used to model particle dynamics. This step encompasses most of the issues related to physics and provides answers to the questions: How do we describe a mechanical system? How do we represent its dynamical evolution? The second step concerns the probabilistic framework that is needed to guide us from particle stochastic models to the statistics of interest in practical situations. This step is more mathematical in nature and provides answers to the questions: What are the main stochastic processes? How do we handle them to stay clear of mathematical pitfalls? To concentrate on the physical issues in the following chapters while relying on a safe probabilistic framework, we give here an outline of the mathematical background.