We have performed a detailed quantitative structural analysis of epitaxial ${[\mathrm{Fe}(3\phantom{\rule{0.3em}{0ex}}\mathrm{nm})∕\mathrm{Cr}(1.2\phantom{\rule{0.3em}{0ex}}\mathrm{nm})]}_{20}$ superlattices by low- and high-angle x-ray diffraction, and energy-filtered transmission electron microscopy on cross-section samples. The interface roughness was changed systematically by varying the substrate temperature (150-250 \ifmmode^\circ\else\textdegree\fi{}C) maintaining all other growth parameters fixed. Direct imaging of the interfaces allows examining the roughness of the individual interfaces and its evolution with thickness. A statistical analysis of the local interface width for the individual layers supplies the roughness static and dynamic exponents. High-temperature samples (250 \ifmmode^\circ\else\textdegree\fi{}C) show roughness decreasing with thickness as a result of surface-diffusion-dominated growth. Low-temperature samples (150 \ifmmode^\circ\else\textdegree\fi{}C) show anomalous non-self-affine roughness characterized by a time-dependent local interface width.