In this colloquium-style talk, I will introduce Sub-Riemannian geometry and highlight one of its central questions: the regularity of its geodesics, known as SR-geodesics. I will focus on our joint work with Figalli, Parusinski, and Rifford, where we prove that on three-dimensional analytic manifolds, all SR-geodesics are C^1. I will outline the main ideas behind the proof and illustrate how a fundamental tool from algebraic geometry—resolution of singularities—plays a key role in this context.