| Algebraic geometry studies the zero locus of polynomial equations connecting the related algebraic and geometrical structures. In several cases, nevertheless the theory is extremely precise and elegant, it is hard to read in a simple way the information behind such structures. A possible way of avoiding this problem is that of associating to polynomials some polyhedral structures that immediately give some of the information connected to the zero locus of the polynomial. In relation to this strategy I will introduce Newton-Okounkov bodies and Tropical Geometry, underlying the connection between the two theories. I will conclude stating a recent result in collaboration with E. Postinghel, where the interplay of tropical geometry and Newton-Okounkov bodies gives a flat degeneration for Mori Dream Spaces. |