Singularities of Algebraic Subvarieties and Problems of Birational Geometry

We consider the connection between the problem of estimating the multiplicity of an algebraic subvariety at a given singular point and the problem of describing birational maps of rationally connected varieties. We describe the method of hypertangent divisors which makes it possible to give bounds for the multiplicities of singular points. The concept of birational rigidity of algebraic varieties is discussed.