I will discuss joint work with Gabriele Di Cerbo on boundedness of Calabi-Yau pairs. Given an elliptically fibered Calabi-Yau manifold, the base of the fibration naturally carries the structure of a Calabi-Yau pair, that is, there exists an effective divisor D on the base, with nice singularities, such that K+D=0. Recent works in the minimal model program suggest that rationally connected Calabi-Yau pairs should satisfy some boundedness properties, that is, they should be parametrized by a finite type scheme. I will show that Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded, if the dimension is less than four. In dimension three, we can actually obtain some more general results, by relaxing some of technical assumptions (joint work in progress with Chen, Di Cerbo, Han, Jiang).