- Two models for the investigation of the quantum damped kicked-rotator problem are introduced and analysed in a unified fashion. For the first model we follow the Caldeira- Leggett approach while the second constitutes a simplification of the Dittrich-Graham model. These models enable one to investigate the effects of noise and dissipation for systems that exhibit chaos in the classical limit and quantum localization otherwise. The rotator is coupled via its angle coordinate to a heat bath that is held at an arbitrary temperature. Noise time- autocorrelations which may arise from such coupling and the validity of the Markovian approximation are discussed.