Two of the most classical topics of study in geometric group theory are mapping class groups and CAT(0) cube complexes. This is in part because they both admit powerful combinatorial-like structures encoding interesting aspects of their geometries: curve graphs for the former and hyperplanes for the later. The broad class of CAT(0) spaces -- while also studied extensively in the literature-- generally lacks an intrinsic combinatorial structure similar to that present in cube complexes or mapping class groups. I will talk about recent work with Petyt and Spriano where we introduce two combinatorial objects for studying CAT(0) spaces: curtains which are analogues of cubical hyperplanes and the curtain model which is a counter part of the curve graph. Such structures allow for vast extensions of theorems known in the above contexts to that of CAT(0) spaces including an Ivanov-stlyle rigidity theorem, a dichotomy of a rank-rigidity flavor and the presence of a universal hyperbolic space for rank-one elements.