Abstract: The study of regularity of the Cauchy-Riemann (d-bar) equation in several complex variables has a long history. In this colloquium, we will present several recent results on the d-bar problem. A common scheme for these solutions is to reproduce differential forms via integral representations. We will explain how the Stein and Rychkov extension operators for function spaces can be used to construct integral formulas. As an application, we will show that the integral representations can be used to study the stability of small deformation of complex structures on domains in a complex manifold.