  Mkn 421 was repeatedly observed with BeppoSAX in 1997-1998. We present
highlights of the results of the thorough temporal and spectral analysis
discussed by Fossati et al. (1999) and Maraschi et al. (1999), focusing on the
flare of April 1998, which was simultaneously observed also at TeV energies.
The detailed study of the flare in different energy bands reveals a few very
important new results: (a) hard photons lag the soft ones by 2-3 ks *a behavior
opposite to what is normally found in High energy peak BL Lacs X-ray spectra*;
(b) the flux decay of the flare can be intrinsically achromatic if a stationary
underlying emission component is present. Moreover the spectral evolution
during the flare has been followed by extracting X-ray spectra on few ks
intervals, allowing to detect for the first time the peak of the synchrotron
component shifting to higher energies during the rising phase, and then
receding. The spectral analysis confirms the delay in the flare at the higher
energies, as above a few keV the spectrum changes only after the peak of the
outburst has occurred. The spectral and temporal information obtained challenge
the simplest models currently adopted for the (synchrotron) emission and most
importantly provide clues on the particle acceleration process. A theoretical
picture accounting for all the observational constraints is discussed, where
electrons are injected at low energies and then progressively accelerated
during the development of the flare.
