Littlewood's conjecture is a famous and long-standing open problem in Diophantine approximation, and is closely related to the action of diagonal matrices on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$ (diagonal action). From this relation and a rigidity property of the action, a breakthrough for this conjecture was made in 2000's. In this talk, I will explain such the relation between Littlewood's conjecture and the diagonal action. I will also explain my result on “quantitative” Littlewood's conjecture which is also led from some properties of the diagonal action.