Architected cellular materials (ACMs) with a periodic micro-structure are often employed in high-performance components obtained through additive manufacturing (AM) technologies due to their high specific strength and stiffness. ACMs are also used in thermal applications, where their high surface-to-mass ratio can be conveniently exploited to enhance heat transfer. In this work, a numerical approach to predict the effective thermal conductivity (ETC) of ACMs obtained by AM is proposed. The model is based on a general numerical homogenisation scheme and an explicit description of the representative volume element (RVE) of the ACM. Numerical analyses have been conducted on 31 RVEs geometries: results show that the macroscopic ETC of ACMs strongly depends on the relative density and the geometrical features of the RVE. Moreover, starting from the database of RVEs geometries, seven configurations are chosen to design graded ACMs through a computer-aided design-compatible topology optimisation method based on non-uniform rational basis spline hyper-surfaces to represent the pseudo-density field, and on the well-known solid isotropic material with penalisation (SIMP) approach. In particular, the penalisation law used in the SIMP method is replaced by a physically-based penalisation scheme obtained by interpolating the results of the homogenisation for each RVE topology and a suitable post-processing phase is developed to recover the distribution of the graded ACM over the structure from the results of the optimisation process. The effectiveness of the proposed approach is shown on 2D and 3D benchmark problems taken from the literature< Leer menos