The paper presents a space-time localized adjoint method (LAM) approximation for the advection-diffusion transport equation. The formulation is based on a space-time discretization in which specialized test functions are defined. These functions locally satisfy the homogeneous adjoint equation within each element. The formulation leads to a general approximation that subsumes many specific methods based on combined Lagrangian and Eulerian approaches, so-called characteristic methods (CM's). The authors refer to the method as an Eulerian-Lagrangian localized adjoint method (ELLAM). The ELLAM approach not only provides a unification of CM methods, but also provides a systematic framework for incorporation of boundary conditions in CM approximations. Example calculations were presented to demonstrate that the ELLAM procedure can handle all types of boundary conditions.