  Context. The main element of the observing program of the
Spectrum-Roentgen-Gamma orbital observatory is a four-year all-sky survey, in
the course of which the entire sky will be scanned eight times.
  Aims. We analyze the statistical properties of AGN and QSOs that are expected
to be detected in the course of the eROSITA all-sky survey (eRASS).
  Methods. According to the currently planned survey strategy and based on the
parameters of the Galactic and extragalactic X-ray background as well as on the
results of the recent calculations of the eROSITA instrumental background, we
computed a sensitivity map of the eRASS. Using the best available
redshift-dependent AGN X-ray luminosity function (XLF), we computed various
characteristics of the eRASS AGN sample, such as their luminosity- and redshift
distributions, and the brightness distributions of their optical counterparts.
  Results. After four years of the survey, a sky-average sensitivity of
~1x10^(-14) erg/s/cm^2 will be achieved in the 0.5-2.0keV band. With this
sensitivity, eROSITA is expected to detect ~3 million AGN on the extragalactic
sky (|b|>10deg). The median redshift of the eRASS AGN will be z~1 with ~40% of
the objects in the z=1-2 redshift range. About 10^4 - 10^5 AGN are predicted
beyond redshift z=3 and about 2 000 - 30 000 AGN beyond redshift z=4, the exact
numbers depend on the poorly known behavior of the AGN XLF in the high-redshift
and luminosity regimes. Of the detected AGN, the brightest 10% will be detected
with more than ~38 counts per PSF HEW, while the faintest 10% will have fewer
than ~9 counts. The optical counterparts of ~95% of the AGN will be brighter
than I_(AB)=22.5mag. The planned scanning strategy will allow one to search for
transient events on a timescale of half a year and a few hours with a
0.5-2.0keV sensitivity of ~2x10^(-14) to ~2x10^(-13) erg/s/cm^2, respectively.
