The goal of this talk is to study positivity of line bundles through the lenses of Newton-Okounkov polygons. This talk will explain the interesting connection that seems to arise between Algebraic Geometry and Convex Geometry. As an application we will show how to read the moving Seshadri constants from the Newton-Okounkov polygons and discuss a few interesting applications. I hope I will have time to discuss an interesting application to syzygies on abelian surfaces as a generalization of the famous Reider's theorem. This is a joint work with Alex Kuronya.