Abstract: Building on work of Okounkov from the 1990s, in 2008 Kaveh and Khovanskii, Lazarsfeld and Mustata showed how to associate to an n-dimensional algebraic variety X and a line bundle a convex body in n-dimensional Euclidean space, the Newton-Okounkov body. In the first part of this talk we will revise construction and main properties of these bodies. In the second part of the talk we will see that for Mori dream spaces, Newton-Okounkov bodies are particularly nice and give rise to toric degenerations (using Anderson’s construction). This is joint work with Stefano Urbinati.