Fused deposition modeling (FDM) also called fused filament fabrication (FFF) is the most used additive manufacturing (AM) technology. The growing impact of AM is due to its various advantages and its applicability to many domains. Many research works have focused on the improvements of FDM technique and optimization of the mechanical properties in order to fabricate parts that can be used in realm industrial applications. In the present work, a review of materials used in the FDM process is proposed together with an analysis of the key parameters affecting their mechanical behaviors. In this framework, FDM materials have been classified into three groups: standard, composite, and smart materials. Previous works have clearly shown that the process parameters have a greater influence on parts made with standard materials than on those made with composites. The effect of the process parameters such as air gap, layer thickness, build orientation, raster orientation, and contour number is discussed in regard to the mechanical solicitations: tensile and compression, three- or four-point bending tests and fatigue. The impact of these process parameters on different material categories is also analyzed. This reveals the specificities related to each materials group. This work can be considered like a global insight for the comprehension of the process—structure—mechanical property relations of common FDM materials. Synthetic remarks and recommendations are proposed for future investigations.