The adjoint state method is a fundamental technique arising in the numerical treatment of inverse and control problems when the underlying simulation is based on finite-differences. Although explicit finite-difference simulation has a number of attractive features, computation of the adjoint of the linearized map is tedious, and naive designs can lead to grossly inefficient code. The purpose of this talk is to present a straightforward derivation of the adjoint state method that eases its implementation in computer code, and to discuss the use of a "checkpointing" scheme that allows the adjoint state equation to be solved efficiently and automatically. I will also discuss the design of a computer package that requires only that the user code a single step of the finite-difference simulation, along with the linearization and its adjoint for that single step; the result is a routine that computes the entire simulation and its linearization and its adjoint.