Abstract: The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation. We shall present the classical methid, and give an approachable introduction to Kim's method. I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html