Let $K$ be a number field, and let $A$ be an abelian variety over $K.$ In order to find the proper setting of the Sato-Tate conjecture concerning the equidistribution of Frobenius elements in the representation of the Galois group of $K$ on the Tate module of $A,$ one of the attempts is the introduction of the algebraic Sato-Tate group $AST_K(A).$ Maximal compact subgroups of $AST_K(A)({\mathbb C})$ are expected to be the key tool for the statement of the Sato-Tate conjecture for $A.$ Following an idea of J-P. Serre, the explicit construction of $AST_K(A)$ will be presented based on P. Deligne's motivic category of absolute Hodge cycles and the corresponding motivic Galois group. I will also discuss the arithmetic properties of $AST_K(A)$ along with explicit computations of $AST_K(A)$ for some families of abelian varieties. This construction of algebraic Sato-Tate group can be done more generally than for abelian varieties, it can also be done for absolute Hodge cycle motives. This is joint work with Kiran Kedlaya.