The distribution of rational points on algebraic varieties is a central problem in number theory. An even more general problem
is to investigate rational points near manifolds, where the algebraic condition is replaced with the non-vanishing curvature
condition. In this talk, we will establish a sharp asymptotic formula for the number of rational points of a given height and
within a given distance to a hypersurface. This has surprising applications to counting rational points lying on the manifold;
indeed setting the distance to zero, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally
stated for projective varieties) in this general setup. In the second half of the talk, we will focus on metric diophantine
approximation on manifolds and its connection with the counting problem described above. A long standing conjecture in this
area is the Generalized Baker-Schmidt Problem, a beautiful Zero versus Infinity law for the Hausdorff measure of
well-approximable points on the manifold in terms of an arbitrary approximation function. As another consequence of the main
counting result above, we settle this problem for all hypersurfaces with non-vanishing Gaussian curvatures. Finally, if time
permits, we will briefly elaborate on the main ideas behind the proof