Topology of algebraic varieties: a long still-ongoing story

In the first part I will survey the study of topological properties of algebraic varieties through three crucial steps: -the seminal work of S. Lefschetz; -the introduction of Hodge theory -the extension of the theory to singular varieties and to maps. Through the theory of toric varieties, I will illustrate how the progress made at each step yields a deeper insight on the combinatorics of convex polytopes. In the second part I will discuss some of the main techniques used in the field: 1. cohomological dimension of affine varietes, 2. semisimplicity of monodromy representations, 3. the yoga of weights .