Log Fano varieties and their applications

Fano varieties play an important role in algebraic geometry. In modern birational geometry, it is more convenient to work with pairs variety + boundary divisor on it. Using this language, various generalisations of Fano varieties were defined: log Fano varieties, varieties of Fano type, etc. These notions appeared to be useful for many problems in birational geometry, and also they are interesting in themselves. We present some results that show how the geometrical properties (e.g. toricity or rationality) of a log Fano variety are connected with the geometry of its boundary. We also discuss applications to the study of the Cremona group.