The Sato–Tate conjecture, originally stated for elliptic curves on 1963, 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 (which is a refinement of the Mumford–Tate conjecture) 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.