SLE curves are introduced by Oded Schramm as the candidates of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points.  Third we introduce Imaginary Geometry and from there give the idea of the proof of the dimension results.