In 1969 Artin and Mazur defined the etale homotopy type Et(X) of scheme X, as a way to homotopically realize the etale topos of X. In the talk I shall present for a map of schemes X-> S a relative version of this notion. We denoted this construction by Et(X/S) and call it the homotopy type of X over S.

It turns out that the relative Homotopy type, can be especially useful in studying the sections of the map X-> S.

In the special case where S = Spec K is the spectrum of a field, the set of sections is just the set of rational points X(K) and then the relative homotopy type Et(X/Spec K) can be used to define obstructions to the existence of a rational point on X.

When K in a number fields it turns out that most known obstructions for the existence of rational points (such as Grothendieck´s section obstruction, the regular and etale Brauer-Manin obstructions, etc.. ) can be obtained in this way and this point a view can be used to show new properties of these obstructions.

In the case where K is a general field or ring this method allows one to get new obstructions that generalized the obstructions above.