Reconfigurable intelligent surfaces (RISs) constitute a promising performance enhancement for next-generation (NG) wireless networks in terms of enhancing both their spectrum efficiency (SE) and energy efficiency (EE). We conceive a system for serving paired power-domain non-orthogonal multiple access (NOMA) users by designing the passive beamforming weights at the RISs. In an effort to evaluate the network performance, we first derive the best-case and worst-case of new channel statistics for characterizing the effective channel gains. Then, we derive the best-case and worst-case of our closed-form expressions derived both for the outage probability and for the ergodic rate of the prioritized user. For gleaning further insights, we investigate both the diversity orders of the outage probability and the high-signal-to-noise (SNR) slopes of the ergodic rate. We also derive both the SE and EE of the proposed network. Our analytical results demonstrate that the base station (BS)-user links have almost no impact on the diversity orders attained when the number of RISs is high enough. Numerical results are provided for confirming that: i) the high-SNR slope of the RIS-aided network is one; ii) the proposed RIS-aided NOMA network has superior network performance compared to its orthogonal counterpart.