A downlink single-input single-output non-orthogonal multiple access setting is considered, in which a base station (BS) is communicating with two legitimate users in two possible scenarios of unsecure environments: existence of an external eavesdropper and communicating through an untrusted relay. For the first scenario, a number of trusted cooperative half-duplex relays is employed to assist with the BS's transmission and secure its signals from the external eavesdropper. Various relaying schemes are proposed and analyzed for that matter: cooperative jamming, decode-and-forward, and amplify-and-forward. For each scheme, secure beamforming signals are devised at the relays to maximize the achievable secrecy rate regions. For the second scenario, with the untrusted relay, achievable secrecy rate regions are derived for two different relaying schemes: compress-and-forward and amplify-and-forward, under two different modes of operation. In the first mode, coined passive user mode, the users receive signals from both the BS and the untrusted relay, and combine them to decode their messages. In the second mode, coined active user mode, the users transmit a cooperative jamming signal simultaneously with the BS's transmission to further confuse the relay. Focusing on half-duplex nodes, the users cannot receive the BS's signal while jamming the relay, i.e., while being active, and rely only on the signals forwarded to them by the relay. It is shown that the best relaying scheme highly depends on the system parameters, in particular distances between the nodes, and also on which part of the secrecy rate region the system is to operate at.