Abstract: The Hodge conjecture, with rational coefficients, states that for a smooth projective variety over \mathbb{C} the image of the cycle class map from Chow groups to Betti cohomology is the group of Hodge classes. Although the integral version of the conjecture is false, Rosenschon and Srinivas proved that the étale version of the integral Hodge conjecture, i.e. using Lichtenbaum cohomology groups instead of Chow groups, is equivalent to the Hodge conjecture with rational coefficients. The goal of this talk is to give an overview of étale motivic cohomology and the Hodge conjecture, followed by a revisit to some of the counter-examples to the integral Hodge conjecture, such as the ones of Atiyah-Hirzebruch, Kollar and Benoist-Ottem, from an étale motivic point of view.