The course will begin with a brief survey of the theory of triangulations of a convex n-gon, which appear in several different places in mathematics. This course will focus on one such, possibly unexpected, appearance, namely in the theory of rigid motions of points in the plane. With this perspective, we will move to non-convex arrangements of points, and discuss the appropriate replacement of a triangulation. Finally, we will say a word about how these constructions are related to representation theory, via configuration spaces of points in the plane and a certain category of quiver representations. I will also mention some open questions in this direction.