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.
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