I will review some results which relate these areas of mathematics. Newton polyhedra theory connects algebraic geometry and the theory of singularities to the geometry of convex polyhedra with integral vertices in the framework of toric geometry. The theory of Newton-Okounkov bodies relates algebra, singularities and geometry of convex bodies outside of that framework. An intermediate version of these theories provides such a relation in the framework of spherical varieties. Those relations are useful in many directions and suggest new unexpected results.