  Aims: This work presents a high--precision variability survey in the field of
the old, super metal-rich open cluster NGC 6791.
  Methods: The data sample consists of more than 75,000 high-precision CCD time
series measurements in the V band obtained mainly at the Canada-France-Hawaii
Telescope, with additional data from S. Pedro Martir and Loiano observatories,
over a time span of ten nights. The field covers an area of 42x28 arcmin^2.
  Results: We have discovered 260 new variables and re-determined periods and
amplitudes of 70 known variable stars. By means of a photometric evaluation of
the membership in NGC 6791, and a preliminary membership based on the proper
motions, we give a full description of the variable content of the cluster and
surrounding field in the range 16<V<23.5. Accurate periods can be given for the
variables with P<4.0 d, while for ones with longer periods the limited
time-baseline hampered precise determinations. We categorized the entire sample
as follows: 6 pulsating, 3 irregular, 3 cataclysmic, 89 rotational variables
and 61 eclipsing systems; moreover, we detected 168 candidate variables for
which we cannot give a variability class since their periods are much longer
than our time baseline.
  Conclusions: On the basis of photometric considerations, and of the positions
of the stars with respect to the center of the cluster, we inferred that 11 new
variable stars are likely members of the cluster, for 22 stars the membership
is doubtful and 137 are likely non-members. We also detected an outburst of
about 3 mag in the light curve of a very faint blue star belonging to the
cluster and we suggest that this star could be a new U Gem (dwarf nova)
cataclysmic variable.
