Complex algebraic varieties are defined by systems of polynomial equations over the field of complex numbers. Their geometry has been extensively studied over the years. The 1 dimensional case corresponds to Riemann surfaces. In dimension 2 we have the theory algebraic surfaces which was understood by the Italian school of Algebraic Geometry at the beginning of the 20th century. The Minimal Model Program aims to generalize these results to higher dimensions. The 3 dimensional case was understood in the 1980s by celebrated work of Mori and others. In this talk I will discuss recent developments on the classification of algebraic varieties in all dimensions.

(This talk is of an introductory nature and does not require previous knowledge of the minimal model program.)