Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. Consequently, studying the size of sets of numbers that can be approximated to within a given precision is very important. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. From a measure-theoretic point of view, this classical set of $\psi$-well-approximable numbers in Euclidean space is well understood. However, it is far less clear what happens when this set, or its various generalisations or higher dimensional analogues, is intersected with other natural sets such as curves, manifolds, or fractals. Studying such questions has become a hugely popular topic of interest in Diophantine Approximation in recent years. In this talk I will discuss recent joint work with Baowei Wang (HUST, China) concerning the particular topic of weighted Diophantine Approximation on manifolds.