A simple mathematical analysis using the shadow method is used to predict the shape, disposition, and spacing of moiré fringes produced by two parallel-line grids. The method is extended to grids of regularly spaced closed-nested-plane figures and then to repeated cell structures. Generalizing, it is shown that for any species of grid “moiré magnification” results. The linear and angular displacements of the fringes are related to the relative rotation and displacement of the grids to show the applicability of moiré fringes as a sensitive tool for positioning or for measuring very small displacements and rotations. A brief discussion is presented of the use of the phenomenon as a visual aid in the classroom.