This article investigates the exploitation of semantic communications in multi-user networks. We propose a novel heterogeneous semantic and bit multi-user framework for providing flawless, customized, and intelligent information transmission. We discuss both orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) for the proposed heterogeneous framework, with an emphasis on investigating the attractive interplay between semantic communications and NOMA, namely NOMA enabled semantic communications and semantic communications enhanced NOMA. 1) For NOMA enabled semantic communications, we propose a semi-NOMA scheme for efficiently facilitating the heterogeneous semantic and bit multi-user communication, which unifies conventional NOMA and OMA schemes. The fundamental performance limit, namely semantic-versus-bit rate region, is characterized, which shows the superiority of the proposed semi-NOMA. 2) For semantic communications enhanced NOMA, we propose an opportunistic semantic and bit communication approach to alleviate the early-late rate disparity issue in NOMA. Numerical case studies demonstrate that significant performance gain can be achieved for NOMA by employing semantic communications than bit communications. Finally, several open research directions are highlighted.