We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations, including the characteristic NIPG, OBB, IIPG, and SIPG schemes. The derived schemes possess combined advantages of Eulerian-Lagrangian methods and discontinuous Galerkin methods. An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG, IIPG, and SIPG scheme. Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG, OBB, IIPG, and SIPG schemes in the context of advection-diffusion equations.