Resonator mirrors with reflectance greater than ~40% are required if free-electron laser (FEL) oscillators are to be extended to wavelengths below 100 nm. The principle of total-external reflection has been used to design multifacet mirrors of several metal films that surpass this requirement between 10 and 100 nm. These reflectors have high reflectance over a relatively broad band which suits well the inherent FEL wavelength tunability. The practical limitations to achieving and maintaining high reflectance include oxide- and carbon-contamination, scattering, and surface-plasmon absorption.