Abstract: In this talk, I will present recent developments on random geometric graphs (RGGs) where data are sampled from low-dimensional manifolds corrupted by noise, with motivations arising from manifold learning and spectral clustering. We establish the convergence of random geometric graphs to weighted Laplace–Beltrami operators and identify critical assumptions on the choice of radius. Building on these theoretical results, we provide practical guidance for constructing RGGs and explore their applications in manifold learning and spectral clustering, particularly when the data lie intrinsically on complex geometric structures such as manifolds.