<p>The aim of this study is to study nonlinear damped torsional vibration of single-walled carbon nanotubes (SWCNTs) surrounded by nonlinear viscoelastic medium under thermal stresses. The novelty of this paper is accounting for nonlinear damping of viscoelastic medium. It means that not only the geometrical nonlinearity is considered but also the nonlinearity raised from viscoelastic medium has been considered and effect of the nonlinear damping of viscoelastic medium on torsional response and frequencies has been included. Nonlocal Eringen’s elasticity as a continuum theory has been adapted to consider the small size effect. The nonlinear strain derived using Green–Lagrange strain relation. Hamilton’s principle conducted to derive equation of motion and boundary conditions. Multiple time scale and Galerkin’s methods have been employed to obtain the nonlinear torsional frequencies and responses. The effect of nonlinear damping and stiffness raised by the viscoelastic medium, nanotube length and diameter, thermal stresses, vibration amplitude, nonlocal scale coefficient and boundary conditions on nonlinear torsional vibration are studied. Results reveal that natural torsional frequencies decrease by increasing damping coefficient of viscoelastic medium at both high and low temperatures. However, nonlinear torsional frequencies increase by increasing elastic medium stiffness and vibration amplitude at both high and low temperatures. Temperature influence on frequencies depends on damping and stiffness of viscoelastic medium, length and diameter of nanotube, and nonlocal scale coefficient. From the results obtained, it is observed that the nonlinear torsional frequencies decrease at as the temperature increase for both Clamped–Clamped (C–C) and Clamped-Free (C-F) nanotubes for all modes of vibration. This study provides important insights into how natural frequencies and corresponding responses are influenced by nonlinear viscoelastic medium and geometrical nonlinearity. As nonlinear damping coefficient of viscoelastic medium increases, the nonlinear torsional frequencies decreases. However, higher values of nonlinear frequency obtained when the value of viscus damping decreases, this is true for all vibration modes. By increasing the diameter, the torsional stiffness increases and leads to an increase torsional vibration which causes response to be damped faster.</p>