The Extreme-ultraviolet Variability Experiment (EVE; see Woods et al., 2009) obtains continuous EUV spectra of the Sun viewed as a star. Its primary objective is the characterization of solar spectral irradiance, but its sensitivity and stability make it extremely interesting for observations of variability on time scales down to the limit imposed by its basic 10 s sample interval. In this paper we characterize the Doppler sensitivity of the EVE data. We find that the 30.4 nm line of He ii has a random Doppler error below 0.001 nm (1 pm, better than 10 km s-1 as a redshift), with ample stability to detect the orbital motion of its satellite, the Solar Dynamics Observatory (SDO). Solar flares also displace the spectrum, both because of Doppler shifts and because of EVE's optical layout, which (as with a slitless spectrograph) confuses position and wavelength. As a flare develops, the centroid of the line displays variations that reflect Doppler shifts and therefore flare dynamics. For the impulsive phase of the flare SOL2010-06-12, we find the line centroid to have a redshift of 16.8 ± 5.9 km s-1 relative to that of the flare gradual phase (statistical errors only). We find also that high-temperature lines, such as Fe xxiv 19.2 nm, have well-determined Doppler components for major flares, with decreasing apparent blueshifts as expected from chromospheric evaporation flows.