The Gan-Gross-Prasad conjecture concerns a restriction or branching problem in the representation theory of real or p-adic Lie groups. It also has a global counterpart which is concerned with a family of period integrals of automorphic forms and the special values of their L-functions. It was proposed by Gross-Prasad for the special orthogonal groups, and then extended by Gan-Gross-Prasad to all classical cases. The Gan-Gross-Prasad conjectures have been established for the tempered representations due to a lot of works, such as Gross-Prasad, Waldspurger, Waldspurger-Moeglin, Raphael Beuzart-Plessis, Gan-Ichino, Wei Zhang, Hang Xue and so on. Recently, they formulate the conjectures for the non-tempered representations of Arthur type. In this talk, we will give a brief introduction to the Gan-Gross-Prasad conjectures and then give several examples to verify the Gan-Gross-Prasad conjectures in the non-tempered cases.