We propose MeshOn, a method that finds physically and semantically realistic compositions of two input meshes. Given an accessory, a base mesh with a user-defined target region, and optional text strings for both meshes, MeshOn uses a multi-step optimization framework to realistically fit the meshes onto each other while preventing intersections. We initialize the shapes' rigid configuration via a structured alignment scheme using Vision-to-Language Models, which we then optimize using a combination of attractive geometric losses, and a physics-inspired barrier loss that prevents surface intersections. We then obtain a final deformation of the object, assisted by a diffusion prior. Our method successfully fits accessories of various materials over a breadth of target regions, and is designed to fit directly into existing digital artist workflows. We demonstrate the robustness and accuracy of our pipeline by comparing it with generative approaches and traditional registration algorithms.