I will present an overview of some explicit methods in the context of the Manin-Mumford and Mordell Conjecture. I will give several specific examples (done with R. Pengo or F. Veneziano) of families of curves whose rational and torsion points can be explicitly determined. I intend to explain some of the techniques that we use, among other there are some results in Galois representation, in the geometry of numbers and in arithmetic geometry. I will avoid the details trying to keep the talk understandable to a large audience.