Boundedness theorems for fibered varieties with trivial canonical bundle

I will explain ideas and techniques behind recent new results showing that several classes of fibered varieties with trivial canonical bundle are bounded, that is, they are parametrised by finitely many families of deformations. Namely, I will show how abelian or K3 fibered Calabi—Yau varieties are bounded, up to simple birational equivalences, in the algebraic category, and how the same results holds in the analytic category for Lagrangian holomorphic symplectic varieties. This is joint work with Engel, Filipazzi, Greer, Mauri.