  This is an expository survey with two goals.
  1) The primary goal is to discuss and highlight the impact of two recent
influential ideas in geometric group theory. The first of which is the notion
of an injective metric space which is a rich class of spaces that was imported
to geometric group theory by Lang and have shown to be of a great effect. The
second is Behrstock-Hagen-Sisto's cubical approximation theorem which provides
a novel and particularly successful approach for studying mapping class groups
(of finite type surfaces) and more generally, hierarchically hyperbolic groups.
  2) Our second goal is to demonstrate how numerous geodesic metric spaces
including hyperbolic spaces, CAT(0) spaces, and hierarchically hyperbolic
spaces admit a strikingly rich equivariant wall structure: a discovery that was
inspired by the aforementioned machines; the cubical approximation theorem as
well as injective metric spaces.
