This paper investigates the physical layer security of non-orthogonal multiple access (NOMA) in large-scale networks with invoking stochastic geometry. Both single-antenna and multiple-antenna aided transmission scenarios are considered, where the base station (BS) communicates with randomly distributed NOMA users. In the single-antenna scenario, we adopt a protected zone around the BS to establish an eavesdropper-exclusion area with the aid of careful channel-ordering of the NOMA users. In the multiple-antenna scenario, artificial noise is generated at the BS for further improving the security of a beamforming-aided system. In order to characterize the secrecy performance, we derive new exact expressions of the security outage probability for both single-antenna and multiple-antenna aided scenarios. To obtain further insights, 1) for the single antenna scenario, we perform secrecy diversity order analysis of the selected user pair. The analytical results derived demonstrate that the secrecy diversity order is determined by the specific user having the worse channel condition among the selected user pair; and 2) for the multiple-antenna scenario, we derive the asymptotic secrecy outage probability, when the number of transmit antennas tends to infinity. Monte Carlo simulations are provided for verifying the analytical results derived and to show that: i)~The security performance of the NOMA networks can be improved by invoking the protected zone and by generating artificial noise at the BS; and ii)~The asymptotic secrecy outage probability is close to the exact secrecy outage probability.