The Busemann-Petty problem: generalizations and modifications

The Busemann-Petty asks whether symmetric convex bodies in R^n with smaller (n-1)-dimensional volume of all central hyperplane sections necessarily have smaller n-dimensional volume. The solution
was completed at the end of the 90's, and the answer is affirmative only in dimensions n\le 4. In this talk we present an overview of the solution, generalizations and applications.