The local Gross-Prasad conjecture is a refinement of the multiplicity one theorem for spherical pairs of Bessel type defined by a pair of special orthogonal groups. The conjecture shows that there is exactly one representation having multiplicity equal to one in each Vogan packet (with generic parameter) and it also depicts this unique representation with an epsilon character. I will introduce some recent progress for the conjecture over R and C, part of the work was joint with Z. Luo. This local conjecture is a necessary ingredient for the global Gross-Prasad conjecture. Besides, the codimension-one case of the conjecture is closely related to the branching problem for special orthogonal groups.