Investigation of fusion reaction mechanism is one of important topics in recent years. In comparison with radioactive ion beam, the beam intensities of weakly bound nuclei are orders of magnitude higher. The study of the reaction mechanism induced weakly bound nuclei can further explore the coupling effect of breakup, transfer and other reaction channels on the fusion process. A lot of experimental data have shown that there are many interesting phenomena in the fusion reaction induced by weakly bound nuclei at energies near the Coulomb barrier, such as "enhancement below the Coulomb barrier'' and "suppression above the Coulomb barrier'' of the complete fusion cross section. In this paper, we mainly review the researches of the suppression phenomenon and discuss the possible reasons for the suppression. The main reason for the suppression phenomenon of the complete fusion cross section is that the weakly bound nuclei break up before entering the fusion barrier, thus reducing the incident flux of the complete fusion reaction channel. At the same time, the experimental results show that the degree of suppression may be related to the mass number and structure of target nuclei. There are three kinds of methods to measure the fusion reaction induced by weakly bound nuclei, which are $\gamma$
ray measurement, charged particle measurement and charged particle -$\gamma$
ray coincidence measurement. The charged particle -$\gamma$
ray coincidence measurement has obvious advantages in reaction channel identification. This paper introduces the three measurement methods and the researches using these three methods at home and abroad, including the researches of our groups. In addition, the recent theoretical research work on fusion reactions induced by weakly bound nuclei is also introduced.