The demands for high data rates in 6G networks have driven the transition toward higher frequencies and larger antenna apertures, giving rise to the near-field communications. In the near-field region, spherical waves enable beam focusing to enhance the received power. However, high-frequency focused beams are highly susceptible to ubiquitous obstacles due to rectilinear trajectories. Particularly in multi-user communications with hybrid precoding, focused beamforming suffers from impaired spectral efficiency under potential multi-user link blockages. In this paper, we propose an Airy beamforming enabled multi-user transmission scheme. The near-field Airy wavefront with a bending trajectory is first developed to cope with the obstructed channels, possessing the dual capability of bypassing obstacles and concentrating energy. Moreover, a low-complexity Airy beamforming enabled multi-user communication scheme is designed. Specifically, Airy beams capable of circumventing obstacles and aligning with users are first obtained through hierarchical Airy beam training. Then, the selected Airy beams are leveraged to configure the analog beamformer to achieve multi-user obstacle-avoiding access without full channel state information acquisition. Finally, the digital beamformer is utilized to further mitigate inter-user interference. In simulations, the beam patterns demonstrate that the proposed Airy beamforming successfully circumvents blockages and aligns with multiple users. Across typical mmWave to THz bands, the proposed scheme outperforms conventional focused beamforming in terms of spectral efficiency.