Abstract-This paper considers the antenna selection (AS) problem for a MIMO non-orthogonal multiple access (NOMA) system. In particular, we develop new computationally-efficient AS algorithms for two commonly-used scenarios: NOMA with fixed power allocation (F-NOMA) and NOMA with cognitive radio-inspired power allocation (CR-NOMA), respectively. For the F-NOMA scenario, a new max-max-max antenna selection (A 3 -AS) scheme is firstly proposed to maximize the system sumrate. This is achieved by selecting one antenna at the base station (BS) and corresponding best receive antenna at each user that maximizes the channel gain of the resulting strong user. To improve the user fairness, a new max-min-max antenna selection (AIA-AS) scheme is subsequently developed, in which we jointly select one transmit antenna at BS and corresponding best receive antennas at users to maximize the channel gain of the resulting weak user. For the CR-NOMA scenario, we propose another new antenna selection algorithm, termed maximumchannel-gain-based antenna selection (MCG-AS), to maximize the achievable rate of the secondary user, under the condition that the primary user's quality-of-service requirement is satisfied. The asymptotic closed-form expressions of the average sumrate for A 3 -AS and AIA-AS and that of the average rate of the secondary user for MCG-AS are derived, respectively. Numerical results demonstrate that the AIA-AS provides better user-fairness, while the A 3 -AS achieves a near-optimal sum-rate in F-NOMA systems. For the CR-NOMA scenario, MCG-AS achieves a near-optimal performance in a wide signal-to-noiseratio regime. Furthermore, all the proposed AS algorithms yield a significant computational complexity reduction, compared to exhaustive search-based counterparts.Index Terms-Multiple-input multiple-output (MIMO), nonorthogonal multiple access (NOMA), antenna selection (AS).