We review the design of the spectrometers included on the Extreme Ultraviolet Explorer (EUVE) astronomical satellite. The spectrometer design is novel, consisting of three variable line space reflecting gratings mounted in a slitless configuration behind a grazing incidence telescope. A collimator is employed to reduce diffuse background radiation to negligible lev-els. Prototype gratings have been mechanically ruled and the best samples recover over 80% of the theoretical efficiency of perfectly formed grooves, reaching 30% absolute at 114Å. Groove profile measurements made using a Talystep profilometer and electron micrographs are compared, and found to agree with the blaze angles derived from reflectivity measure-ments. In the final design, grating blaze angles have been optimized to minimize second order contamination. Prototype collimator EUV measurements show peak transmission of 90% of theoretical, with transmission outside the main lobe of less than 0.5%, and a scattering level less than 10-4arcmin-1. Using measured performance characteristics of the collimator, telescope, gratings and detector, we have determined the sensitivity of the instrument for a 40,000 sec observation; the average 3-σ sensitivity of continuum flux is approximately 2x10-27erg/cm2/sec/Hz. This is a factor of 100 dimmer than a bright known EUV source, and is comparable to the sensitivity of the all-sky survey which will be carried out on the EUVE.