This paper presents the state-of-the-art of additive manufacturing of composites for processing functional, load-bearing components. A general overview of different additive manufacturing methods is provided, and specific attention is focused on fused filament fabrication-based composites processing. Different process modeling strategies are summarized, and key aspects of these models are discussed. Significant results such as thermal and fluid flow characteristics, effects of nozzle geometry on melt flow, fiber orientation, bead spreading, and solidification, the formation of residual stresses, and deformation behavior are discussed from computational modeling perspective. The scientific advancement, model limitations, and future modeling needs are prescribed reviewing the current works. A general overview of material development in nano-micro-macro-scale reinforcement is also presented. Different length-scales of reinforcement has its own challenges and promises. The continuous fiber reinforcement has a great potential for being the next-generation composites manufacturing technology. However, the challenges in reducing the void content, better bonding between the fiber–matrix, and layer-layer adhesion, and process uncertainty are some of the key areas yet to advance. Based on the current limitations on computational modeling, materials development, and process modeling studies, future research needs and recommendations are provided.