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.