Composite cylindrical shells are widely used as structural elements in modern engineering applications like aircraft, ships, and industrial structures. However, most of the applications of laminated composite shells use polymer as a matrix depicting viscoelastic behaviour. Thus, it is important to account for the viscoelastic behaviour in the study of nonlinear dynamics of composite shells. Permoon et al. analysed the nonlinear vibration response of fractional Kelvin- Voigt isotropic viscoelastic cylindrical shells using Love’s thin shell theory, Galerkin method and the method of multiple scales. The linear transient vibration response of viscoelastic composite cylindrical shells is investigated in time domain using Newmark average acceleration method by Yi et al. To the best of the authors’ knowledge, the nonlinear transient and steady-state response of laminated composite cylindrical shells have not been investigated in the literature. In the present study, the nonlinear transient/periodic response of viscoelastic cylindrical shells is investigated using the generalised Maxwell model, FSDT based finite element and Newmark’s direct time integration method. The detailed parametric study is carried out depicting the interplay of stiffness and damping for different lamination schemes and boundary conditions.