  This is a survey paper dealing with holomorphic foliations, with emphasis on
residue theory and its applications. We start recalling the definition of
holomorphic foliations as a subsheaf of the tangent sheaf of a manifold. The
theory of Characteristic Classes of vector bundles is approached from this
perspective. We define Chern classes of holomorphic foliations using the
Chern-Weil theory and we remark that the Baum-Bott residue is a great tool that
help us to classify some foliations. We present throughout the survey several
recent results and advances in residue theory. We finish by presenting some
applications of residues to solve for example the Poincar\'e problem and the
existence of minimal sets for foliations.
