On the boundedness of Calabi-Yau varieties in low dimension

I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele Di Cerbo.
Recent work in the minimal model program suggests that pairs with trivial log canonical 
class should satisfy some boundedness properties. 
I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are 
indeed log birationally bounded. This implies birational boundedness of elliptically fibered 
Calabi-Yau manifolds with a section, in dimension up to 5.
If time allows, I will also try to discuss a first approach towards boundedness of rationally
connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and, C. Jiang).