In this talk, we give an overview of space-time mixed norm estimates for dispersive equations on noncompact symmetric spaces. We focus on two key estimates: the Strichartz inequality and the Smoothing property. The former is based on the pointwise kernel estimate, while the latter relies on the Stein-Weiss inequality, also known as the Hardy-Littlewood-Sobolev inequality with double weights. These results differ from the classical ones in the Euclidean setting due to the particular geometry at infinity. Additionally, we discuss how the geometric properties of the discrete group affect these results on locally symmetric spaces.