Abstract
We present two recent developments in the approximation theory of modules. The first one investigates boundaries of this theory, namely the classes naturally occurring in homological algebra, but not providing for approximations (e.g., the class of all flat Mittag-Leffler modules).
We introduce the key tools for their study which involve set-theoretic methods combined with (infinite dimensional) tilting theory. The second development concerns tilting classes, their structure over commutative rings, and the recent generalization to silting modules and classes.