1. Introduction

The purpose of this note is to survey some recent results in higher-dimensional birational geometry. A glance to the table of contents may give the reader some idea of the topics that will be treated. I have attempted to give an informal presentation of the main ideas, emphasizing the common grounds, addressing a general audience. In §3, I could not resist discussing some details that perhaps only the expert will care about, but hopefully will also introduce the non-expert reader to a subtle subject.

Perhaps the most significant trend in Mori theory today is the increasing use, more or less explicit, of the logarithmic theory. Let me take this opportunity to advertise the Utah book [Ko], which contains all the recent software on log minimal models. Our notation is taken from there.

I have kept the bibliography to a minimum and made no attempt to give proper credit for many results. The reader who wishes to know more about the results or their history could start from the references listed here and the literature quoted in those references.

The end of a proof or the absence of it will be denoted with a ▫ . Anyway here proof always means “proof“: a bare outline will be given at best, usually only a brief account of some of what the author considers to be the main ideas.