Appears in collection : Summer School 2019 - Foliations and algebraic geometry

The purpose of this course is to present the basics of the general theory of (singular) holomorphic foliations. We will begin with the general definition of a (regular) foliation and its relation with Frobenius Theorem. We will then introduce the singular analogues of these notions in the holomorphic setting and with some emphasis on the case of foliations of dimension 1 and foliations of codimension 1. These definitions will be illustrated with natural examples arising in the projective plane (space). Next, the fundamental notions of lead ans of lead holonomy sill be discussed and examples will be given.All the preciding stems from the regular part of the foliations so, at this point, we will also discuss the nature of the singular points and, in particular, the case of hyperbolic singularities. When the ambient manifold is of dimension 2, then a lot can be said and, in this direction, we will state Seidenberg's reduction Theorem. The course will end with the special case of foliations that are transverse to a fibration and related constructions.