  We present a regularization strategy that leads to well-conditioned boundary
integral equation formulations of Helmholtz equations with impedance boundary
conditions in two-dimensional Lipschitz domains. We consider both the case of
classical impedance boundary conditions, as well as the case of transmission
impedance conditions wherein the impedances are certain coercive operators. The
latter type of problems is instrumental in the speed up of the convergence of
Domain Decomposition Methods for Helmholtz problems. Our regularized
formulations use as unknowns the Dirichlet traces of the solution on the
boundary of the domain. Taking advantage of the increased regularity of the
unknowns in our formulations, we show through a variety of numerical results
that a graded-mesh based Nystr\"om discretization of these regularized
formulations leads to efficient and accurate solutions of interior and exterior
Helmholtz problems with impedance boundary conditions.
