The redshift (z) and Early Universe Spectrometer (ZEUS) is an echelle grating spectrometer designed to study the history of star formation in the Universe from about 2 billion years after the Big Bang to the present by observing submillimeter and far-infrared spectral lines from distant dusty galaxies. ZEUS has moderate resolving power (R~1000), and large spectral coverage so as to optimize extragalactic point source sensitivity in the telluric submillimeter (350, 450, and 610 um) windows. When completed, ZEUS will have a 4 x 64-element array of TES PUD bolometers delivering an instantaneous 64-element spectrum for each of 4 spatial positions on the sky. ZEUS is designed for use on the 15 m JCMT telescope on Mauna Kea. We also plan to use it on the 12 m APEX telescope at the Chajnantor site in northern Chile. Our scientific goals include (1) investigating star formation in the early Universe by measuring the redshifted fine-structure lines from distant (z ~1 to 4) (proto-) galaxies, (2) measuring the redshifts of optically obscured submillimeter galaxies by detecting their bright 158 um [CII] line emission, and (3) investigating the properties of starburst and ultraluminous galaxies in the local Universe by observing their [CI] and mid-J CO rotational line emission.