Compact Hyperkaehler varieties: from smooth to singular

Compact Hyperkaeler manifolds play a central role in complex algebraic geometry, as they arise as one of the building blocks of compact Kaehler manifolds with trivial canonical class. I will briefly recall the (very rich) theory of compact Hyperkaehler manifolds, hence I will pass to their analogue in the singular setting, where several different definitions are possible and convenient for different purposes. I will discuss the relation among these definitions and present some problem arising in the different singular settings, discussing in particular some problem I have been recently interested in.