A general framework for the multi-scale topology optimisation (TO) of lattice structures (LSs) is presented in this work. The proposed method involves: Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field describing the LS representative volume element (RVE) topology, the Solid Isotropic Material with Penalisation (SIMP) approach and the strain energy-based homogenisation method (SEHM) to perform the scale transition. The main contributions of this work are essentially three. Firstly, physical responses are defined at different scales and their gradient is evaluated by exploiting the NURBS local support property and the Dirichlet’s problem properties at the RVE scale. Secondly, the computational efficiency of the SEHM based on elements strain energy over that of the SEHM based on elements averaged stresses is rigorously proven. Finally, to show the effectiveness of the method, numerical analyses are conducted on 2D and 3D problems. A sensitivity analysis of the optimised topology to the integer parameters of the NURBS hyper-surface is carried out. Moreover, the influence of the initial guess and of the macroscopic loading condition on the RVE optimised topology is investigated. The minimum length-scale requirement is also integrated into the problem formulation as a manufacturing constraint.Read less <