In this talk we will discuss recent results on the classification of complex projective algebraic varieties i.e. compactifications of subsets of

**C**^n defined as the zeros of polynomial equations. Two varieties are birationally isomorphic if they admit the same meromorphic functions, or equivalently if they have isomorphic open subsets in the Zariski topology (that is, they agree outside a subset of smaller dimension).The minimal model program allows us to choose a "simplest" representative in each birational equivalence class. These representatives are known as canonical models. They are either positively, trivially or negatively curved and there has been much recent progress in understanding their moduli spaces.