Fused deposition modelling (FDM) has attracted researchers’ interest in myriads of applications. The enhancement of its part using fillers to print nanocomposites is a cutting-edge domain of research. Industrial acceptance is still a challenge, and researchers are investigating different nanofillers and polymer matrix combinations to investigate FDM-printed nanocomposites. Carbon nanotubes, graphene, and cellulose are heavily studied nanofillers because of their astonishing properties, biocompatibility, and ability to tailor the final performance of the FDM-printed nanocomposite part. This work presents a comprehensive review of polymer nanocomposites based on these nanofillers. Important examples, case studies, and results are discussed and compared to elaborate the understanding of the processing of nanocomposites, filaments, printing, and the characterisation of these nanocomposites. A detailed and exhaustive discussion of the prospective computational models, with challenges and a future road map, is provided, enabling the scientific community to understand these nanocomposites and their FDM processing for wider industrial applications and acceptance.