The Chabauty-Coleman method is a powerful tool for the explicit computation of rational points on higher-genus curves. However, it is limited to curves whose Jacobians have Mordell–Weil rank less than the genus of the curve. To extend its applicability, Kim introduced a significant generalisation of the method - now known as the Chabauty-Kim method - which has led to effective results for a variety of new examples. In this talk, we will give a brief overview of the Chabauty-Kim method and discuss recent developments in its effective application to rational points on hyperelliptic curves.