Extreme events are defined as those events in which the characteristics (e.g. field strength, speed, intensity of radiation, energies) of the associated phenomena (e.g. solar flares, coronal mass ejections, solar proton events) are some orders of magnitude larger than in other events. Such strong events commonly occur about two years before and after sunspot maximum and some strong events occur as well in the declining phase before the solar activity minimum [Bothmer V., Zhukov A. The 11 Sun as the prime source of space weather, in: Bothmer, V., Daglis, I. (Eds.), Space Weather: Physics and Effects, Springer Praxis Books, 12 pp. 438, 2007]. In the first part of the paper the characteristics of the Jan. 2005 and Dec. 2006 events are given. This is followed by a presentation of the effects that were encountered on technological systems and also addresses the issue of what could have occurred on biological systems during such events. The second part of the paper deals with how one should go about analyzing solar extreme events - as part of the global distribution of all events or as "outliers" with their own special characteristics. © 2008 COSPAR.