In this paper we review recent numerical and theoretical developments of particle self-assembly on fluid and elastic membranes and compare them to available experimental realizations. We discuss the problem and its applications in biology and materials science, and give an overview of numerical models and strategies to study these systems across all length-scales. As this is a very broad field, this review focuses exclusively on surface-driven aggregation of nanoparticles that are at least one order of magnitude larger than the surface thickness and are adsorbed onto it. In this regime, all chemical details of the surface can be ignored in favor of a coarse-grained representation, and the collective behavior of many particles can be monitored and analyzed. We review the existing literature on how the mechanical properties and the geometry of the surface affect the structure of the particle aggregates and how these can drive shape deformation on the surface.