Purity and approximation theory influence a wide range of areas in mathematics including model theory of modules, silting and tilting theory, cotorsion pairs and abelian model structures and many more. Moreover, related classification problems in abelian and triangulated categories are strongly connected with spaces such as Ziegler spectra and prime spectra. We aim to bring together experts and young researchers with an interest in these topics to share their different viewpoints and approaches. The workshop will include five mini lecture series introducing a range of fundamental topics, and contributed talks.