  Since its commissioning in fall 1996, the stereoscopic system of Imaging
Atmospheric Cherenkov Telescope (IACTs) of HEGRA with an energy threshold of
500 GeV, an angular resolution of 0.1 degree and an energy resolution of 20%
per individual photon, and an energy flux sensitivity nu F_nu at 1 TeV of
1E-11erg/cm**2 s (S/N = 5 sigma) for one hour of observation time has been used
to monitor the BL Lac object Mkn 421 on a regular basis. In this letter, we
report detailed temporal and spectral information about the TeV characteristics
of Mkn 421 in 1997 and 1998. We study the light curve, the shortest time scales
of flux variability, the differential spectra on diurnal basis for several days
with good gamma-ray statistics and the time averaged energy spectrum. Special
emphasis will be put on presenting the data taken during the world-wide April
1997 multiwavelength campaign. We compare the Mkn 421 results with the results
obtained for the BL Lac object Mkn 501 and discuss possible implications for
the emission mechanism and the Diffuse Intergalactic Background Radiation.
