Non-orthogonal Multiple Access (NOMA) has been recognised as a promising technology for next generation wireless networks. NOMA theories use a new domain, power domain, for multiple access techniques and this enables new solutions in many areas of the future networks. This new research trend, despite being very promising, raises many challenges for researchers in recent years. Some interesting questions include how to utilize the power to improve the security of the system and how to efficiently harvest the energy from the received data signals for later use in order that we may have a green network with more environment-friendly devices. Meanwhile, Mobile Edge Computing (MEC) spreads the computational ability to the edge of the networks, allows more computations closer to the end users ,and results in lower latency, more secure transmission and other advantages. With MEC, NP-hard problems in wireless networks, such as spectrum allocation and energy consumption minimization problems, can be solved in a practical way. The overall goal of this thesis is to propose new techniques to utilize the power domain in NOMA-based networks to improve their security as well as enable energy harvesting capacity and smart computing at the edge of the network. This will be achieved by investigating new system models for MEC-based NOMA networks as well as developing, implementing and examining new techniques in the mentioned topics.