The development of carbon nanotube(CNT)-polymer composites advocates for a better understanding of their physical and mechanical properties that depend on the diameter of the embedded CNTs. Given that the experimental assessment of size effects is extremely difficult, the use of numerical models can be enormously helpful. However, since size effects might be observed both at the nano- and the macroscale, an adequate multiscale procedure is required. In this thesis, numerical techniques are explored to develop a multiscale approach for the analysis of size effects in the elastic response of CNT-polymer composites. Atomistic simulations, such a molecular mechanics and molecular dynamics, are used for the characterization of the composites and their components at the nanoscale. The obtained results are then used to investigate size effects in the macroscopic properties of CNT-polymer composites using continuum models and efficient finite element techniques. Molecular mechanics simulations on tensile carbon nanotubes show that their axial stiffness and axial strain field depend on the CNT diameter. Moreover, it is found that the axial strain field can be accurately reproduced using nonlocal continuum models if optimal nonlocal parameters, that vary with the nanotube diameter, and a suitable nonlocal kernel are used. Although the numerical solution of nonlocal problems is typically challenging, higher order B-spline finite elements overcome the issues encountered when standard approximation techniques are employed. Further, molecular dynamics simulations on CNT-polymer composites show that the CNT diameter alters the atomic structure and the mechanical properties of the ordered layer of polymer chains forming around the nanotube —the interphase. Such a layer has a significant impact on the mechanical properties of the composite. Although the role of the nanotubes during elastic deformation of the composite is negligible due to the weak nonbonded interface interactions, the interphase–thanks to its highly ordered atomic structure–is shown to enhance its mechanical properties. Here, molecular mechanics simulations at the nanoscale and the numerical solution of an equivalent continuum model at the macroscale indicate that the composite stiffness increases when the diameter of the carbon nanotubes is decreased. When possible, the reliability of the results in this thesis has been assessed by means of analytical models and experimental or numerical results in the literature. Therefore, this study proposes a computational framework to improve our understanding of the mechanical response of CNT-polymer composites and the size effects on their elastic properties.@en