We introduce Project MEGaSaURA: the Magellan Evolution of Galaxies Spectroscopic and Ultraviolet Reference Atlas. MEGaSaURA comprises medium-resolution, rest-frame ultraviolet spectroscopy of N=15 bright gravitationally lensed galaxies at redshifts of 1.68<z<3.6, obtained with the MagE spectrograph on the Magellan telescopes. The spectra cover the observed-frame wavelength range 3200<{lambda}_o_<8280{AA}; the average spectral resolving power is R=3300. The median spectrum has a signal-to- noise ratio (S/N)=21 per resolution element at 5000{AA}. As such, the MEGaSaURA spectra have superior S/N and wavelength coverage compared to what COS/HST provides for starburst galaxies in the local universe. This paper describes the sample, the observations, and the data reduction. We compare the measured redshifts for the stars, the ionized gas as traced by nebular lines, and the neutral gas as traced by absorption lines; we find the expected bulk outflow of the neutral gas, and no systemic offset between the redshifts measured from nebular lines and the redshifts measured from the stellar continuum.