Abstract: The Sato-Tate conjecture, originally stated for elliptic curves without complex multiplication, predicts the equidistribution of the normalized Frobenius traces with respect to the Sato-Tate measure, given by the pushforward of the Haar measure on SU(2). We would like to work on an analogous question for abelian varieties of dimension g > 1; the generalized Sato-Tate conjecture, introduced by Serre, which predicts the equidistribution on a certain compact Lie group: the Sato-Tate group. In 1966, Serre presented remarkable links between the Mumford-Tate group and the Sato-Tate group. Thus, the algebraic Sato-Tate group appears as an intermediate group between the Mumford-Tate group and the Sato-Tate group. Indeed, if the algebraic Sato-Tate conjecture holds for some particular abelian variety A of dimension g > 1, we can obtain the Sato-Tate group and try to deduce some new instances of the generalized Sato-Tate conjecture. The main goal of this talk is to present new results in the direction of the algebraic Sato-Tate conjecture, building on the previous work of Serre, Kedlaya and Banaszak.