Abstract: Nondivergence estimates for flows on homogeneous spaces were first applied to Diophantine approximation in the 1990s, when conjectures of Baker and Sprindzhuk on (strong) extremality of nondegenerate manifolds were proved. Recently similar methods were developed for handling generalizations of those conjectures, dealing with the set-up of systems of linear forms. I will review the methods, explain how those more general problems are motivated, and state some of the results. Parts of this work are joint with Victor Beresnevich, Gergory Margulis and Junbo Wang.