In this talk, we investigate the computation of barycentres for generic optimal transport costs, extending beyond the well-studied Wasserstein barycentres. We introduce a fixed-point algorithm that generalizes prior works and prove its convergence under minimal assumptions. Our approach applies to a broad class of cost functions and probability measures, providing a flexible framework for the numerical computation of  generic transport barycentres. Joint work with Eloi Tanguy and Nathaël Gozlan.