Failure of the integral Hodge conjecture for hypersurfaces

The integral Hodge conjecture relates algebraic cycles on smooth complex projective varieties to their topological and analytic aspects. After Atiyah and Hirzebruch found the first counterexample, the integral Hodge conjecture is studied as a property that a given variety might satisfy or not. I will first give an overview of known results for different classes of varieties. Then I will focus on hypersurfaces, which are, due to Kollár, the first non-torsion counterexamples to the integral Hodge conjecture. I want to present Kollár's arguments in more detail and explain how they can be combined in order to fully determine the cokernel of the cycle class map for many hypersurfaces.