The authors describe a method of local adaptive grid refinement for the solution of the steady Euler equations in two dimensions, which automatically selects regions requiring mesh refinement by measuring the local truncation error. Our method of refinement uses locally uniform fine rectangles which are superimposed on a global coarse grid. Possibly several nested levels of refined grids will be used until a given accuracy is attained. The fine grid patches are in the same coordinate system as the underlying coarse grid. All the data management is done in the computational plane, where, since we use rectangular grids, the data structures and bookkeeping can be very simple.