We present a science forecast for the extended Baryon Oscillation Spectroscopic Survey (eBOSS) survey. Focusing on discrete tracers, we forecast the expected accuracy of the baryonic acoustic oscillation (BAO), the redshift-space distortion (RSD) measurements, the fNL parameter quantifying the primordial non-Gaussianity, the dark energy and modified gravity parameters. We also use the line-of-sight clustering in the Lyman a forest to constrain the total neutrino mass. We find that eBOSS luminous red galaxies, emission line galaxies and clustering quasars can achieve a precision of 1, 2.2 and 1.6 per cent, respectively, for spherically averaged BAO distance measurements. Using the same samples, the constraint on fob is expected to be 2.5, 3.3 and 2.8 per cent, respectively. For primordial non-Gaussianity, eBOSS alone can reach an accuracy of a (f(NL)) similar to 10-15. eBOSS can at most improve the dark energy figure of merit by a factor of 3 for the Chevallier-Polarski-Linder parametrization, and can well constrain three eigenmodes for the general equation-of-state parameter. eBOSS can also significantly improve constraints on modified gravity parameters by providing the RSD information, which is highly complementary to constraints obtained from weak lensing measurements. A principal component analysis shows that eBOSS can measure the eigenmodes of the effective Newton's constant to 2 per cent precision; this is a factor of 10 improvement over that achievable without eBOSS. Finally, we derive the eBOSS constraint (combined with Planck, Dark Energy Survey and BOSS) on the total neutrino mass, sigma (Em(upsilon)) = 0.03 eV (68 per cent CL), which in principle makes it possible to distinguish between the two scenarios of neutrino mass hierarchies.