A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with geometric genus 0 and the boundary of the moduli space of deformations of the surface. The goal of this thesis is to analyze this relation for Godeaux surfaces. To do this, first, we give a description of some boundary components of the moduli space of Godeaux surfaces. Second, we explicitly construct certain exceptional vector bundles of rank 2 on Godeaux surfaces, stable with respect to the canonical class. Finally, we examine the relation between such boundary components and exceptional vector bundles of rank two on Godeaux surfaces.