<p>The free vibration characteristics of a shallow-spherical shell mounted on the viscoelastic foundation which is characterized by the fractional-order Maxwell model is studied. Firstly, the fractional-order Maxwell viscoelastic model of foundation is introduced, this model is applicable to engineering scenarios where the foundation exhibits significant viscoelastic behavior, such as the clay stratum and liquid saturated porous stratum, and the frequency-dependent complex modulus and the reaction force resulted from foundation are given. Then, with the help of the stress function, the governing differential equations of the shallow-spherical shell mounted on the viscoelastic foundation are derived in term of only the stress function and the transversal displacement. The complete analytic solution of the stress function and the transversal displacement are finally obtained in term of the classical Bessel functions and the modified Bessel functions. The complex natural frequency is introduced to reflect the attenuation feature of the free vibration. Finally, the numerical example is provided for shallow-spherical shell with the fixed-edge boundary conditions. The first three orders of natural frequencies and the vibration pattern and the corresponding attenuation index are estimated and shown in tables and 3D cloud figure. The influences of the viscoelastic foundation and the two typical vibration patterns with the axisymmetric and the non-axisymmetric properties are discussed in detail.</p>