The influence of the breakup channel on the elastic scattering and on the fusion process for systems that involve weakly bound stable nuclei has been widely investigated in the last years. One of the most used approaches to address these problems is to investigate the energy dependence of the interaction nuclear potential. For tightly bound systems at energies close to the Coulomb barrier, this energy dependence exhibits the behavior known as threshold anomaly. On the other hand, for various weakly bound systems the nuclear potential as a function of energy has a completely different behavior known as breakup threshold anomaly. In this report we present an overview of this subject and we discuss the different experimental measurements and theoretical interpretations that have been published. In this framework, we present the results of elastic scattering cross sections in the ^6,7Li+⁸⁰Se systems that have been recently measured at the TANDAR laboratory and we discuss the conclusions obtained from these studies.