We describe the results of a search for transit-like events caused by giant planets occulting stars in the old, metal-rich open cluster NGC 6791 based on BVI-colour photometry from eight nights of observations with the 2.54 m Nordic Optical Telescope. To extract the light curves we have used both PSF photometry (DAOPHOT) and the difference imaging technique (ISIS). We have re-analyzed observations from earlier campaigns to search for multiple transits, determine periods of long-period variables, and detect eclipsing binaries. We confirm 20 known variables and have discovered 22 new low-amplitude variables with amplitudes in the range 7-40 mmag and periods 1-16 days. We have found the primary and secondary eclipses of two eclipsing binaries based on the new and older photometric campaigns. The search for transit-like events has turned up a few single-transit candidates. The transit depths are 10 mmag in both the V and I filters over periods of 1.0-2.5 hours, but future observations are required to see whether identical transit-like events recur in these same stars. Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.