Abstract: This paper introduces a novel computational approach for level-set based topology optimisation of hyperelastic materials at large strains This, to date, is considered an unresolved open problem in topology optimisation due to its extremely challenging nature Two computational strategies have been proposed to address this problem The first strategy resorts to an arc-length in the pre-buckling region of intermediate topology optimisation (TO) iterations where numerical difficulties arise (associated with nucleation, disconnected elements, etc), and is then continued by a novel regularisation technique in the post-buckling region In the second strategy, the regularisation technique is used for the entire loading process at each TO iteration The success of both rests on the combination of three distinct key ingredients First, the nonlinear equilibrium equations of motion are solved in a consistent incrementally linearised fashion by splitting the design load into a number of load increments Second, the resulting linearised tangent elasticity tensor is stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator Third, and with the purpose of avoiding excessive numerical stabilisation, a scalar degradation function is applied on the regularised linearised elasticity tensor, based on a novel regularisation indicator field The robustness and applicability of this new methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications Crucially, the performance of all the designs has been tested at a post-processing stage without adding any source of artificial stiffness Specifically, an arc-length Newton-Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10− 12