Toric varieties and Calabi-Yau complete intersections in them can be realised as vacuum configurations of gauged linear sigma models (GLSMs) with abelian gauge groups. Interesting phenomena and new types of geometries occur when one considers more general settings. From a GLSM perspective, this includes strong coupling effects in GLSMs with non-abelian gauge groups, new dualities, and geometries arising via non-perturbative effects. The geometries one finds can be complete intersections in non-toric spaces, non-complete intersections, or non-commutative resolutions of singular spaces. The aim of this talk is to give an example-based overview of these new phenomena with a focus on the challenges that occur compared to toric settings.