  We give an introductory account of two recent approaches towards an effective
proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter
method, which is usually called the method of Chabauty--Kim or non-abelian
Chabauty in the literature, has the advantage that in some cases it has been
turned into an effective method to determine the set of rational points on a
curve, and we illustrate this by presenting three new examples of modular
curves where this set can be determined.
