Abstract: In this talk I will discuss recent progress around the variation of the Mordell-Weil rank in families of elliptic curves. The first part will be dedicated to give a thorough introduction to the theme, by motivating and discussing the state of the art. The second part will be dedicated to presenting a recent work joint with Dan Loughran (Bath). More precisely, let X be surface endowed with a (non-constant) elliptic fibration with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly larger than the latter. I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre.