The problem of finding rational or integer solutions to polynomial equations is one of the oldest problems in mathematics and is one of the key driving forces in the development of Number Theory. In the last 15 years new methods were developed that can sometimes effectively solve this problem. These methods attempt to find the solutions inside the larger set of solutions of the same equation in the field of p-adic numbers as the vanishing set of some computable function. When these methods work they give the rational solutions to arbitrarily large p-adic precision, which usually suffices to rigorously recover the full set of solutions.
I will survey the new methods, originating from the work of Kim and from the more recent work of Lawrence and Venkatesh. I will then explain my work with Muller and Srinivasan that uses a p-adic version of the notion of norms on line bundles and associated heights, as used for example in arithmetic dynamics, to give a new approach to some Kim type results.