Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena as they are intimately related with localization, both at the level of module categories and of derived categories.
In these notes, we review the main features of infinite dimensional tilting modules. We discuss the relationship with approximation theory and with localization. Finally, we focus on some classification results and we give a geometric interpretation of tilting.