Resolution of Singularities in algebraic geometry

The title may sound too much of an old issue. However  recently in the case of positive characteristic cases we found new phenomena and interesting techniques much more than the case of characteristic zero. For this rare precious opportunity, I do not want  to spend much time on  the history of the topic nor about possible applications. I will focus my effort to explain all the essence of the proofs in positive characteristics and arithmetic cases. If my talk becomes too technical to any one among the audience, please stop me and question me. Any critics and advises will be most welcome.  If any time left near the end of my talk I will make some comments on history of the problem and possible future applications.