The non-abelian Chabauty method of Minhyong Kim is one of the premier methods for computing rational or integral points on curves in practice. This course will give a thorough description of the foundations of the method and the theoretical ingredients it combines, and will conclude with an overview of some computations of rational or integral points.
A course summary can be found here.
These are copies of the notes I used for the lectures. Please feel free to contact me via email if you have any comments, queries or corrections. The eventual aim is to write these up properly into a book -- watch this space!