  The goal of this survey is to discuss the Quantitative non-Divergence
estimate on the space of lattices and present a selection of its applications.
The topics covered include extremal manifolds, Khintchine-Groshev type
theorems, rational points lying close to manifolds and badly approximable
points on manifolds. The main emphasis is on the role of the Quantitative
non-Divergence estimate in the aforementioned topics within the theory of
Diophantine approximation, and therefore this paper should not be regarded as a
comprehensive overview of the area.
