This article clarifies pivotal security issues in the context of non-orthogonal multiple access (NOMA) technology and presents potential methodologies to address them at the physical layer. Specifically, we first clarify the channel conditions for an external eavesdropper to use successive interference cancellation (SIC) for cancelling interference, which has recently been considered in the literature of physical layer security for NOMA. We then explicitly demonstrate the advantages and drawbacks of using NOMA to achieve wireless communication security; that is, NOMA improves the security of a strong user, but reduces the security of a weak user. Finally, we discuss a strategy to protect the confidentiality of a weak user's information from a strong user, while still allowing the latter to seek SIC for harnessing the benefits of NOMA. Research challenges and future works along the aforementioned three different aspects of security issues in NOMA are presented in this article.