In this paper, we propose a multi-beam non-orthogonal multiple access (NOMA) scheme for hybrid millimeter wave (mmWave) systems and study its resource allocation. A beam splitting technique is designed to generate multiple analog beams to serve multiple users for NOMA transmission. Compared to conventional mmWave orthogonal multiple access (mmWave-OMA) schemes, the proposed scheme can serve more than one user on each radio frequency (RF) chain. Besides, in contrast to the recently proposed single-beam mmWave-NOMA scheme which can only serve multiple NOMA users within the same beam, the proposed scheme can perform NOMA transmission for the users with an arbitrary angle-of-departure (AOD) distribution. This provides a higher flexibility for applying NOMA in mmWave communications and thus can efficiently exploit the potential multi-user diversity. Then, we design a suboptimal two-stage resource allocation for maximizing the system sum-rate. In the first stage, assuming that only analog beamforming is available, a user grouping and antenna allocation algorithm is proposed to maximize the conditional system sum-rate based on the coalition formation game theory. In the second stage, with the zero-forcing (ZF) digital precoder, a suboptimal solution is devised to solve a non-convex power allocation optimization problem for the maximization of the system sum-rate which takes into account the quality of service (QoS) constraint. Simulation results show that our designed resource allocation can achieve a close-to-optimal performance in each stage. In addition, we demonstrate that the proposed multi-beam mmWave-NOMA scheme offers a higher spectral efficiency than that of the single-beam mmWave-NOMA and the mmWave-OMA schemes.