The EUV Imaging Spectrometer (EIS) on Hinode will observe solar corona and upper transition region emission lines in the wavelength ranges 170 - 210 Å and 250 - 290 Å. The line centroid positions and profile widths will allow plasma velocities and turbulent or non-thermal line broadenings to be measured. We will derive local plasma temperatures and densities from the line intensities. The spectra will allow accurate determination of differential emission measure and element abundances within a variety of corona and transition region structures. These powerful spectroscopic diagnostics will allow identification and characterization of magnetic reconnection and wave propagation processes in the upper solar atmosphere. We will also directly study the detailed evolution and heating of coronal loops. The EIS instrument incorporates a unique two element, normal incidence design. The optics are coated with optimized multilayer coatings. We have selected highly efficient, backside-illuminated, thinned CCDs. These design features result in an instrument that has significantly greater effective area than previous orbiting EUV spectrographs with typical active region 2 - 5 s exposure times in the brightest lines. EIS can scan a field of 6×8.5 arc min with spatial and velocity scales of 1 arc sec and 25 km s-1 per pixel. The instrument design, its absolute calibration, and performance are described in detail in this paper. EIS will be used along with the Solar Optical Telescope (SOT) and the X-ray Telescope (XRT) for a wide range of studies of the solar atmosphere.