Melt compounding using an extruder is the method for mixing functionalized materials into plastics to add functionality to existing plastics and improve their performance. The performance of the composites depends on the dispersion state of the mixed materials in the composites. Therefore, the analysis and evaluation of the dispersion state of adding materials in the composites are necessary for considering the performance of the composites. In this article, we focus on the melt compounding of composite materials using thermoplastics and outline the analysis and evaluation methods of melt compounding and mixing process of plastics by an extruder, citing the references. First, we explain the concepts of dispersive mixing and distributive one as the basic theory of melt compounding. Next, we introduce the following examples for the analysis and evaluation of melt compounding by an extruder, 1) Dispersion of binary mixtures, 2) Dispersion of functionalized fillers such as nanoclay and vapor-grown carbon fiber, 3) Evaluation by flow simulation of polymer melts in an extruder, 4) Estimation of pigment dispersion and online estimation during melt compounding as the other topics. Finally, we describe future problems and tasks as the conclusion.