Compact hyper-Kähler manifolds and their Lagrangian fibrations are higher-dimensional generalizations of K3 surfaces and their elliptic fibrations. I will present a recent exploration of the geometry of isotrivial Lagrangian fibrations conducted with Y. Kim and R. Laza. We show that the smooth fiber of such a fibration is isogenous to the power of an elliptic curve and present a trichotomy arising from the Kodaira dimension of the minimal Galois cover of the base which trivializes monodromy. We are motivated in part by the search for new deformation types of hyper-Kähler manifolds and the boundedness problem.