Unformatted post-print version of the accepted article There has been considerable interest in the nucleation and crystallization of polymers in the presence of nanoparticles (NPs, or nanofillers in general, NFs). Most of the extensive work in this area has focused on anisotropic, non-Brownian NFs (e.g., clay sheets, carbon nanotubes) whose spatial dispersion state in these nanocomposites is controlled by the process by which they are formed. While NF spatial dispersion is thus generally poor, or poorly characterized, in many works, thermodynamic handles that can be used to control NF dispersion state in the polymer melt include (a) favorable interactions between the polymer chains and the bare NP surfaces, or (b) the density and length of the chains, with the same chemistry as the matrix, grafted to the NP surface. These relatively large NFs merely act as immovable objects that affect the kinetics of nucleation by providing heterogeneous sites, and the crystallization rate by confining the polymer in the melt state. The dispersion state of the NFs can dramatically 2 affect the nucleation and crystallization of the matrix, but in most cases reported, the NFs increase nucleation efficiency relative to the neat polymer. At higher NF loadings, the effect of polymer confinement by the NFs dominates, leading to a decrease in crystal growth rates. In this review, we first describe the most important lessons learned from these commonly studied systems and then use this knowledge to understand the results obtained when small, mobile spherical NPs (typically smaller than 100 nm in size) are used as the nanofillers. The role of NP mobility, which provides for dynamic confinement of the polymer melt, on the kinetics of polymer crystallization (nucleation, growth, and overall crystallization) and how this behavior is mostly consistent with the case of immobile NF is a second important focus of our review. In addition to the role of NFs on crystallization kinetics, we discuss recently reported nanoparticle ordering phenomena, i.e., how the crystallization of polymers under appropriate conditions can move and organize small spherical NPs within the amorphous regions of the semicrystalline morphology. Such phenomena are clearly not observed for large NFs and hence provide a point of departure from past work in this well-traveled area. This work was supported by grants DE-SC0018182, DE-SC0018135, and DE-SC0018111 funded by the U.S Department of Energy, Office of Science. We would like to thank the financial support provided by National Key R&D Program of China (Grant No. 2017YFE0117800) and the National Natural Science Foundation of China (Grant Nos. 21574141, 51820105005, and 52050410327). We also acknowledge the financial support from the BIODEST project; this project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 778092. A.J.M. acknowledges funding from MINECO, project MAT2017-83014-C2-1-P, and from Basque Government through grant IT1309-19. A.A.K. acknowledges funding from the Gates Millennium Scholars program under Grant No. OOP1202023 from the Bill & Melinda Gates Foundation. R.A.P.-C is supported by the China Postdoctoral Science Foundation (Grant No. 2020M670462).