Reconfigurable intelligent surface (RIS) has been regarded as a promising technology since it has ability to create the favorable channel conditions. This paper investigates the secure communications of RIS assisted non-orthogonal multiple access (NOMA) networks, where both external and internal eavesdropping scenarios are taken into consideration. More specifically, novel approximate and asymptotic expressions of secrecy outage probability (SOP) for the k-th legitimate user (LU) are derived by invoking imperfect successive interference cancellation (ipSIC) and perfect successive interference cancellation (pSIC). To characterize the secrecy performance of RIS-NOMA networks, the diversity order of the k-th LU with ipSIC/pSIC is obtained in the high signal-to-noise ratio region. The secrecy system throughput of RIS-NOMA networks is discussed in delay-limited transmission mode. Numerical results are presented to verify theoretical analysis that: i) The SOP of RIS-NOMA networks is superior to that of RIS assisted orthogonal multiple access (OMA) and conventional cooperative communication schemes; ii) As the number of reflecting elements increases, the RIS-NOMA networks are capable of achieving the enhanced secrecy performance; and iii) The RIS-NOMA networks have better secrecy system throughput than that of RIS-OMA networks and conventional cooperative communication schemes.