ABSTRACT The Tayler instability is an important but poorly studied magnetohydrodynamic (MHD) instability that likely operates in stellar interiors. The non-linear saturation of the Tayler instability is poorly understood and has crucial consequences for dynamo action and angular momentum transport in radiative regions of stars. We perform three-dimensional MHD simulations of the Tayler instability in a cylindrical geometry, including strong buoyancy and Coriolis forces as appropriate for its operation in realistic rotating stars. The linear growth of the instability is characterized by a pre-dominantly m = 1 oscillation with growth rates roughly following analytical expectations. The non-linear saturation of the instability appears to be caused by secondary shear instabilities and is also accompanied by a morphological change in the flow. We argue, however, that non-linear saturation likely occurs via other mechanisms in real stars where the separation of scales is larger than those reached by our simulations. We also observe dynamo action via the amplification of the axisymmetric poloidal magnetic field, suggesting that Tayler instability could be important for magnetic field generation and angular momentum transport in the radiative regions of evolving stars.