Grzegorz Banaszak Title: Algebraic Sato-Tate and Sato-Tate conjectures Abstract: Let $K$ be a number field and let $\rho_{l} : G_{K} \rightarrow GL(V_l)$ be a strictly compatible family of $l$-adic representations, according to Serre, associated with a pure, polarized, rational Hodge structure. In the lecture I will introduce Algebraic Sato-Tate and Sato-Tate conjectures in this general framwork. I will explain how these conjectures are related to the motivic approach by Serre to generalize the classical Sato-Tate conjecture. I will also state number of past and recent results concerning these conjectures. This is joint work with Kiran Kedlaya.