Abstract: A connection is a way of differentiating a vector field on a manifold. When the manifold has a complex structure, there is a holomorphic version of this notion which, once mild singularities are permitted, is called a regular (meromorphic) connection. I will explain how a regular connection on a Riemann surface (i.e. surface with a complex structure) is the same thing as an affine surface, roughly a surface made by gluing together polygons in the plane where, crucially, the gluing maps are allowed to rotate and stretch edges. Using this perspective, I will explain why the map from the moduli space of affine surfaces (aka the moduli space of regular connections) that inputs a regular connection and outputs its holonomy homomorphism is a submersion away from the locus of finite-area translation surfaces. I will also explain a similar nonabelian variant of this result that identifies precisely when the monodromy map from the moduli space of affine surfaces to the character variety of homomorphisms from the fundamental group to the group of affine transformations of C is a submersion. This latter result is related to work of Hubbard and Hejhal on affine and projective structures. This work is joint with Matt Bainbridge and Jane Wang.