Non-orthogonal multiple access (NOMA) has been viewed as a potential candidate for the upcoming generation of wireless communication systems. Comparing to traditional orthogonal multiple access (OMA), multiplexing users in the same time-frequency resource block can increase the number of served users and improve the efficiency of the systems in terms of spectral efficiency. Nevertheless, from a security view-point, when multiple users are utilizing the same time-frequency resource, there may be concerns regarding keeping information confidential. In this context, physical layer security (PLS) has been introduced as a supplement of protection to conventional encryption techniques by making use of the random nature of wireless transmission media for ensuring communication secrecy. The recent years have seen significant interests in PLS being applied to NOMA networks. Numerous scenarios have been investigated to assess the security of NOMA systems, including when active and passive eavesdroppers are present, as well as when these systems are combined with relay and reconfigurable intelligent surfaces (RIS). Additionally, the security of the ambient backscatter (AmB)-NOMA systems are other issues that have lately drawn a lot of attention. In this paper, a thorough analysis of the PLS-assisted NOMA systems research state-of-the-art is presented. In this regard, we begin by outlining the foundations of NOMA and PLS, respectively. Following that, we discuss the PLS performances for NOMA systems in four categories depending on the type of the eavesdropper, the existence of relay, RIS, and AmB systems in different conditions. Finally, a thorough explanation of the most recent PLS-assisted NOMA systems is given.