Abstract: In many classical settings, such as hyperbolic geometry, it is known that small deformations of a smooth geometric structure on a manifold are parametrised by small perturbations of the holonomy representation. This is a particular (affine) representation of the fundamental group. Hyperbolic Dehn surgery allows this fact to be extended even to hyperbolic 3-manifolds with certain singularities. In this talk, we will explore this idea in the setting of polyhedral manifolds—manifolds with a metric induced by a Euclidean triangulation. We will introduce the concepts of the developing map, monodromy and holonomy in this setting, via simple examples. We will see that desirable geometric properties can be deduced from the holonomy. And finally, we will think about how to understand the relationship between holonomy and deformations of polyhedral manifolds.