Homogeneous dynamics is a rapidly developing field that investigates the ergodic properties of Lie group actions on its homogeneous spaces. Over the years, it has revealed deep and far-reaching connections with number theory, particularly in the area of Diophantine approximation. In this talk, I will provide an overview of some of the key results in homogeneous dynamics and examine problems in Diophantine approximation from a dynamical perspective. In particular, I will discuss recent joint work with Timothée Bénard and Weikun He, in which we resolve Mahler's 1984 problem concerning Khintchine's theorem for self-similar measures.