This mini-course of three lectures will cover the main theoretic ingredients required to understand Montgomery’s sub-Riemannian theorem on principa lG-bundles. The lectures will also serve a larger purpose to acquaint a general audience to the problems of sub-Riemannian geometry. The exposition will be a blend of control theory, symplectic geometry and Hamiltonian formalism. We will begin with the Orbit theorem and the Max-imum Principle of optimal control, two major cornerstones of sub-Riemannian geometry.We will then revisit classical hyperbolic and elliptic models of non-Euclidean geometry in the spirit of F. Klein’s ”Erlangen program” but diffused with modern perspectives inherited from control theory. We will then link this material to a more general class of sub-Riemannian problems on Lie groups of central importance for the theory of symmetric Riemannian spaces. In the third lecture we will discuss Montgomery’s theorem and its relevance for the problems of applied mathematics.