  This doctoral thesis has two objectives. The first objective is to introduce
a notion of equivalence for singular foliations that preserves their transverse
geometry and is compatible with the notions of Morita equivalence of the
holonomy groupoids and the transverse equivalence for regular foliations that
appeared in the 1980's.
  The second one is to describe the structures behind quotients of singular
foliations and to connect these results with their associated holonomy
groupoids.
  It also wants to give an introduction to the notion of singular foliations as
given by Androulidakis and Skandalis in arXiv:math/0612370, as well as to their
relation with Lie groupoids and Lie algebroids.
