Absolute desingularization in characteristic zero

This paper is an expository lecture notes originally based on a lecture on the results of [Tem1] given by the author at the workshop on Motivic Integration in May 2008, at ICMS, Edinburgh. Since a substantial progress was done since May 2008, it seemed natural to include the new results of [BMT], [Tem2] and [Tem3] in this exposition. We will mainly concentrate on the functorial non-embedded desingularization constructed in [Tem2] because it seems that the results of [Tem3] on the embedded case can be improved further. We pursue expository goals, so we will concentrate on explaining the results and the main ideas of our method and we will refer to the cited papers for proofs and technical details. Also, we try to include more examples and general remarks than in a pure research paper. Thus, this survey can serve as a companion to or a light version of [Tem1] and [Tem2]. I would like to warn the reader that the current situation described in the paper can change soon (similarly to the change since 2008), but this is always a danger with a survey on an active research area.