Topics on singularities and bifurcations of vector fields

These notes are essentially meant to be a continuation of the lecture notes [D]. Results are presented that have mostly been obtained after 1993. In a first section we describe the classification of singularities of smooth vector fields in real 3-space up to codimension 4. Besides giving the description, attention goes to the different techniques that have been used. A second section deals with the study of the unfoldings of planar singularities and how this relates to the study of polynomial Lienard equations and of Abelian integrals. The last section deals with the study of singular perturbations for 2-dimensional vector fields, essentially from a geometric point of view. Throughout the whole text considerable emphasis is put on the one hand on blow up of singularities, and on the other hand on rescaling of families and its generalization: blow up of families.