We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. In this thesis, we use an existing HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control in the high regularity case. We also propose a new HDG method to approximate the solution of a Dirichlet boundary control problem governed by a linear elliptic convection diffusion PDE. Although there are many works in the literature on Dirichlet boundary control problems for the Poisson equation, we are not aware of any existing theoretical or numerical analysis works for convection diffusion Dirichlet control problems. We obtain well-posedness and regularity results for the Dirichlet control problem, and we prove optimal a priori error estimates in 2D for the control in both the high regularity and low regularity cases. As far as the authors are aware, there are no existing comparable results in the literature. Moreover, we present numerical experiments to demonstrate the performance of the HDG methods and illustrate our numerical analysis results.