Topics in sub-Riemannian geometry

This thesis is concerned with three different problems in sub-Riemannian geometry faced during my PhD. The first one is a problem in differential geometry and is about the local conformal classification of a certain class of sub-Riemannian structures. In the second one we deal with topology, and our main result establish some path-fibration properties for the Endpoint map. In the third and last problem, we begin the development of some variational calculus around critical points of the endpoint map, called abnormal controls, and we estabilish a counterpart of the classical Morse deformation techniques and of the Min-Max variational principle.