Abstract
The goal of this survey is to discuss the quantitative nondivergence esti-mate on the space of lattices and present a selection of its applications. The topicscovered include extremal manifolds, Khintchine-Groshev-type theorems, rationalpoints lying close to manifolds, and badly approximable points on manifolds. Themain emphasis is on the role of the quantitative nondivergence estimate in theaforementioned topics within the theory of Diophantine approximation; therefor