Statistical estimation in a geometric context has become an increasingly important issue in recent years, not least due to the development in ML of non-linear dimension reduction techniques, which involve projecting high-dimensional data onto a much lower-dimensional sub-manifold. The search for statistical guarantees justifying both the use and the effectiveness of these algorithms is now a much-studied area. In this talk, we will take a geometric view of the issue, and see how some usual curvature quantities are translated into algorithmic guarantees. First, we will see that upper bounds on sectional curvatures give good properties for barycenter estimation, and then we will see that a lower bound on Ricci curvature implies the existence of depth points, giving rise to robust statistical estimators. Those works are based on joint works with Victor-Emmanuel Brunel (ENSAE Paris) and Shin-ichi Ohta (Osaka University).