The novel concept of near-field non-orthogonal multiple access (NF-NOMA) communications is proposed. By exploiting the analog beamformers focusing on specific locations, the far-to-near successive interference cancellation order can be further facilitated. In the proposed NF-NOMA, the two NOMA users in different angular directions with distinct quality of service (QoS) requirements can be grouped into one cluster and are served by one analog beamformer focusing on multiple locations. To maximize the sum rate of higher QoS (H-QoS) users, the analog beamformer is first designed using the beam-splitting technique, which focuses the energy on both two NOMA users at two different locations. Then, a singular value decomposition based zero-forcing (SVD-ZF) digital beamformer is designed to mitigate the inter-cluster interference. Furthermore, an antenna allocation algorithm is proposed by employing the many-to-one matching method. Finally, an iterative algorithm is proposed to obtain suboptimal power allocation solutions via the fractional programming. Numerical results demonstrate that: i) in contrast to the conventional far-field NOMA, the proposed NF-NOMA schemes can achieve a higher spectral efficiency even if the HQoS users are far located; and ii) NF-NOMA transmission always outperforms near-field orthogonal multiple access transmission.