For X a smooth elliptic K3 surface, it is known that the rank of the Mordell-Weil group can take any value between 0 and 18. In higher dimensions, the existence of a bound on rkMW(X/B) for Calabi-Yau varieties is guaranteed by recent results. Such bounds play an important role in physics. After a general introduction, I will discuss a method that paves the way to determine effective bounds on the rank of the Mordell-Weil group of higher-dimensional elliptic fibrations, and its application to threefolds of Kodaira dimension zero and Calabi-Yau fourfolds.