The stresses in periodic Mo/Si, W/Si, and Mo/C multilayer films were determined from wafer-curvature measurements. The layer thickness of each material was varied systematically, and parametric stress contours were generated, showing contours of constant stress in the two-dimensional layer thickness parameter space. These results illustrate that the net stress in a periodic multilayer is not an intrinsic property of the film (for specific deposition conditions) but, rather, depends strongly on the individual layer thicknesses. X-ray diffraction measurements show (a) how the lattice spacing in the W and Mo crystallites varies with layer thickness, and (b) in the case of the W/Si films, how the phase composition of the polycrystalline W layers vary with W layer thickness. In the case of the W/Si and Mo/Si multilayers, irreversible stress changes were observed after the samples were stored in air at room temperature for a period of several months. Stress-temperature measurements made on the as-deposited W/Si and Mo/Si samples also reveal irreversible stress changes (both positive and negative, depending on the layer thicknesses) after thermal cycling to 300 degrees C; x-ray diffraction measurements were used to identify any associated changes in the W and Mo microstructure. We describe mechanisms that can explain the observed stress behavior, and also discuss the significance of these results, particularly with regard to the use of these films for high-performance multilayer x-ray optics.