Let G/H be a reductive homogeneous space and Γ be a (torsion-free) discrete subgroup of G. The quotient of G/H by the left Γ-action is always a manifold in the Riemannian setting (i.e. when H is compact), but not always otherwise: the Γ-action on G/H needs to be proper. In the last 40 years or so, the study of proper actions on non-Riemannian homogeneous spaces has been one of the fertile areas in mathematics to which various methods (Lie-theoretic / dynamical / homotopical / ...) were applied and produced partially overlapping but different results. In this mini-course, we plan to discuss the following topics in this area:

• • Basics on reductive Lie groups.
• • Properness criterion in terms of the Cartan projection (Kobayashi, Benoist) and its consequences.
• • Proper SL(2,R)-actions vs. proper surface group actions (joint work with Maciej Bocheński).

• Necessary conditions for the existence of proper cocompact actions (joint work with Fanny Kassel and Nicolas Tholozan).

If time permits, we will also discuss:

• • A conjectural picture of proper cocompact actions (Tholozan's geometric fibration conjecture).
• • Relation to Anosov representation theory (Guéritaud-Guichard-Kassel-Wienhard, Kassel-Tholozan, ...)