The world around us appears to involve lengths and angles. From these emerge the classical notions of shape and symmetry configured in 3 dimensions. Our mathematical ancestors realised that these notions should be important, not only for construction and surveying, but also for understanding "life, the universe and everything". In the process of simplifying complicated structures it turns out that an important role is played by conformal geometries -- these are spaces where there is a notion of angle but not length. We will discuss some elegant tools for working with these less than rigid geometries and how they help treat other problems such as compactification. This is the theory behind making non-compact spaces compact by adding points in some appropriate way. There are applications to the representation theory of groups, geometric analysis, and physics.