Theoretically, full-duplex (FD) communications can double the spectral-efficiency (SE) of a wireless link if the problem of self-interference (SI) is completely eliminated. Recent developments towards SI cancellation techniques have allowed to realize the FD communications on low-power transceivers, such as small-cell (SC) base stations. Consequently, the FD technology is being considered as a key enabler of 5G and beyond networks. In the context of 5G, FD communications have been initially investigated in a single SC and then into multiple SC environments. Due to FD operations, a single SC faces residual SI and intra-cell co-channel interference (CCI), whereas multiple SCs face additional inter-cell CCI, which grows with the number of neighboring cells. The surge of interference in the multi-cell environment poses the question of the feasibility of FD communications. In this article, we first review the FD communications in single and multiple SC environments and then provide the state-of-the-art for the CCI mitigation techniques, as well as FD feasibility studies in a multi-cell environment. Further, through numerical simulations, the SE performance gain of the FD communications in ultra-dense massive multiple input multiple-output enabled millimeter wave SCs is presented. Finally, potential open research challenges of multi-cell FD communications are highlighted.