  These lecture notes attempt to invite the reader towards the theory of
singular foliations, both smooth and holomorphic. In addition to a systematic
review of the foundations, and an attempt to put in order examples and several
elementary constructions, we detail several recent tools developed for
non-commutative geometry, in particular the holonomy groupoid of Androulidakis
and Skandalis and various methods for resolving singularities. We also
introduce various homotopic notions, and end with a series of open questions.
