Resolution of singularities in algebraic geometry

Abstract: I will present my way of proving resolution of singularities of an algebraic variety of any dimension over a field of any characteristic. There are some points of general interest, I hope, technically and conceptually more than just the end result. The resolution problem for all arithmetic varieties (meaning algebraic schemes of finite type over the ring of integers) is reduced to the question of how to extend the result from modulo pm to modulo p(m+1) after a resolution of singularities over Q. I will discuss certain problems which arise in this approach.