Elliptic fibrations are geometric spaces fibered by elliptic curves and they play an important role in Mathematics and Physics. In mathematics, they turn out to be good test cases for studying techniques and conjectures on fiber spaces with Calabi-Yau fibers and, in physics, many string theory models are associated to elliptically fibered Calabi-Yau varieties. In this talk, we will discuss some results on the birational geometry of elliptic fibrations, their relationship to string theory and as well as higher dimensional generalizations.