This paper introduces the active simultaneously transmitting and reflecting surface (ASTARS) to assist non-orthogonal multiple access (NOMA) communications, leveraging stochastic geometry to model the spatial positions of pairing users. The authors design independent reflection/transmission phase-shift controllers for ASTARS to align the phases of cascaded channels at pairing users. They derive closed-form and asymptotic expressions for the outage probability and ergodic data rate of ASTARS-NOMA networks in the presence of perfect/imperfect successive interference cancellation (pSIC). The diversity orders and multiplexing gains for ASTARS-NOMA are also derived. The system throughput of ASTARS-NOMA is investigated in both delay-tolerant and delay-limited transmission modes. Numerical results show that ASTARS-NOMA with pSIC outperforms ASTARS-assisted orthogonal multiple access (ASTARS-OMA) in terms of outage probability and ergodic data rate. Increasing the power amplification factors can further reduce the outage probability within a certain range. The system throughputs of ASTARS-NOMA are superior to those of ASTARS-OMA in both delay-limited and delay-tolerant transmission modes.This paper introduces the active simultaneously transmitting and reflecting surface (ASTARS) to assist non-orthogonal multiple access (NOMA) communications, leveraging stochastic geometry to model the spatial positions of pairing users. The authors design independent reflection/transmission phase-shift controllers for ASTARS to align the phases of cascaded channels at pairing users. They derive closed-form and asymptotic expressions for the outage probability and ergodic data rate of ASTARS-NOMA networks in the presence of perfect/imperfect successive interference cancellation (pSIC). The diversity orders and multiplexing gains for ASTARS-NOMA are also derived. The system throughput of ASTARS-NOMA is investigated in both delay-tolerant and delay-limited transmission modes. Numerical results show that ASTARS-NOMA with pSIC outperforms ASTARS-assisted orthogonal multiple access (ASTARS-OMA) in terms of outage probability and ergodic data rate. Increasing the power amplification factors can further reduce the outage probability within a certain range. The system throughputs of ASTARS-NOMA are superior to those of ASTARS-OMA in both delay-limited and delay-tolerant transmission modes.