What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r + s is smaller than the dimension of the ambient space, unless there is some special geometrical relation between them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink, which include very classical conjectures in number theory, rely on this philosophy. In this talk I will give a survey on some of these questions and I will show how some of these results apply to other problems of Diophantine nature.