Lagrangian fibered Hyper-Kaehler manifolds are the natural
generalization of elliptic K3 surfaces. They are examples of
completely integrable system and appear naturally in several areas of
algebraic geometry. Lagrangian fibrations have also been used to study
and construct examples of compact Hyper-Kaehler manifolds and
(possibly singular) symplectic varieties. In this talk I will give an
overview of some old and more recent compactification techniques for
quasi-projective Lagrangian fibrations and applications to the study
of Prym, Intermediate Jacobian, and Albanese fibrations.