I will describe various results on the singular and coherent cohomology of smooth projective varieties, with emphasis on
relatively recent techniques.
I will begin by quickly reviewing more classical aspects of the topology of algebraic varieties and Hodge theory,
together with their applications to important cohomological results like the Kodaira-Nakano vanishing theorem.
Subsequently, I will address more recent theorems on the Betti and Hodge numbers of varieties over the complex numbers
(for instance that birational Calabi-Yau manifolds have the same Hodge numbers). Part of this story is related to p-adic
integration and the Weil conjectures, and will naturally take us through the positive characteristic and finite fields
world as well.
Further techniques that will appear will be chosen from among motivic integration, derived categories, and module theory
over exterior algebras.