Metasurfaces have exhibited considerable capability for generating Airy beams. However, the available plasmonic/dielectric metasurfaces Airy-beam generators have low transmission efficiency and/or poor quality of generated beam because they lack the amplitude modulation. Hyperbolic metamaterials (HMMs) have recently provided an alternative strategy for building high-performance meta-devices that are capable of flexibly modulating the phase, amplitude and polarization state of light. Here we reveal that both the propagation phase and the Pancharatnam–Berry phase can contribute to the local transmission phase of circularly polarized electromagnetic waves by using HMMs. This thus provides us with great freedom to design HMM units with different cross-sections to independently control the transmission phase and amplitude. Here, we design circularly polarized Airy-beam generators in the microwave and near-infrared domains, which require binary phase and polynary amplitude, and validate the good performance in the microwave experiment. Our work can facilate the generation of a complicated light field that highly requires independent and complete control of the transmission phase and amplitude under circularly polarized incidence.