A discontinuous fiber-reinforced polymer composite (DFRPC) provides superior mechanical performances in material extrusion additive manufacturing (MEAM) parts, and thus promotes their implementations in engineering applications. However, the process-induced structural defects of DFRPCs increase the probability of pre-mature failures as the manufactured parts experience complicated external loads. In light of this, the meso-structures of the MEAM parts have been discussed previously, while systematic analyses reviewing the studies of the micro-structural formations of the composites are limited. This paper summarizes the current state-of-the-art in exploring the correlations between the MEAM processes and the associated micro-structures of the produced composites. Experimental studies and numerical analyses including fiber orientation, fiber attrition, and micro-voids are collected and discussed. Based on the review and parametric study results, it is considered that the theories and numerical characterizations on fiber length attrition and micro-porosities within the MEAM-produced composites are in high demand, which is a potential topic for further explorations.