Abstract: Layered structures made of two and more materials with different properties can meet the needs of industrial development. However, the abrupt change of material properties at the interface of the laminated structures can easily cause some interface problems, such as stress concentration, interface cracks, and interface delamination phenomena. Functionally graded materials refer to utilize a continuously changing component gradient instead of the original sudden change interface, which can eliminate or weaken the abrupt change of the physical properties and then increase the bonding strength of the layered structures. In this paper, the research object is the functionally graded multilayered one-dimensional quasicrystal cylindrical shells. By virtue of the pseudo-Stroh formalism and the propagator matrix method, we establish the layered one-dimensional quasicrystal cylindrical shells model with the material parameters following the power-law type distribution along its radius direction, and obtain the exact thermo-electro-elastic solution of the functionally graded layered one-dimensional quasicrystal cylindrical shells with simply supported boundary condition. Numerical examples are carried out to investigate the influences of the exponential factor on temperature, electric, phason and phonon fields of the functionally graded layered one-dimensional quasicrystal cylindrical shells subjected to both inner and outer surfaces temperature variations, especially the effects on physical quantities at the inner and outer surfaces of the layered one-dimensional quasicrystal cylindrical shells. The obtained results indicate that: the exponential factor can change the distribution characteristic of material parameters, which can cause a significant influence on the physical quantities in the temperature, electric, phason, and phonon fields; by increasing the exponential factor, the deformation at the internal surface induced by temperature stimuli is reduced and the strength of the layered one-dimensional quasicrystal cylindrical shells is improved. The results obtained in this paper can provide a reliable theoretical basis for the design and manufacture of functionally graded layered one-dimensional quasicrystal cylindrical shells.