What can we say about the variation of the rank in a family of elliptic curves ? We know in particular that if infinitely many curves in the family have non-zero rank, then the set of rational points is Zariski dense in the associated elliptic surface. We use a “conjectural substitute” for the geometric rank (or rather for its parity) : the root number. For a non-isotrivial family, under two analytic number theory conjectures I show that the root number is -1 (resp. +1) for infinitely many curves in the family. On isotrivial families however, the root number may be constant : I describe its behaviour in this case.