The next generation of wireless systems will take the concept of communications and networking to another level through the seamless integration of terrestrial, aerial, satellite, maritime and underwater communication systems. Reconfigurable intelligent surface (RIS) is an innovative technology which, with its singular features and functionalities, can expedite the realization of this everywhere connectivity. Motivated by the unparalleled properties of this innovatory technology, this article provides a comprehensive discussion on how RIS can contribute to the actualization and proper functioning of future integrated terrestrial/non-terrestrial (INTENT) networks. As a case study, we explore the integration of RIS into non-orthogonal multiple access (NOMA)-based satellite communication networks and demonstrate the performance enhancement achieved by the inclusion of RIS via numerical simulations. Promising directions for future research in this area are set forth at the end of this article.